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In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit-evasion games on the graph, or (in the case of locally finite graphs) as topological ends of topological spaces associated with the graph.

Ends of graphs may be used (via Cayley graphs) to define ends of finitely generated groups. Finitely generated infinite groups have one, two, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition for groups with more than one end.

Definition and characterization

Ends of graphs were defined by Template:Harvs in terms of equivalence classes of infinite paths.^[1] A Template:Visible anchor in an infinite graph is a semi-infinite simple path; that is, it is an infinite sequence of vertices v₀, v₁, v₂, ... in which each vertex appears at most once in the sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph. According to Halin's definition, two rays r₀ and r₁ are equivalent if there is another ray r₂ (not necessarily different from either of the first two rays) that contains infinitely many of the vertices in each of r₀ and r₁. This is an equivalence relation: each ray is equivalent to itself, the definition is symmetric with regard to the ordering of the two rays, and it can be shown to be transitive. Therefore, it partitions the set of all rays into equivalence classes, and Halin defined an end as one of these equivalence classes.

An alternative definition of the same equivalence relation has also been used:^[2] two rays r₀ and r₁ are equivalent if there is no finite set X of vertices that separates infinitely many vertices of r₀ from infinitely many vertices of r₁. This is equivalent to Halin's definition: if the ray r₂ from Halin's definition exists, then any separator must contain infinitely many points of r₂ and therefore cannot be finite, and conversely if r₂ does not exist then a path that alternates as many times as possible between r₀ and r₁ must form the desired finite separator.

Ends also have a more concrete characterization in terms of havens, functions that describe evasion strategies for pursuit-evasion games on a graph G.^[3] In the game in question, a robber is trying to evade a set of policemen by moving from vertex to vertex along the edges of G. The police have helicopters and therefore do not need to follow the edges; however the robber can see the police coming and can choose where to move next before the helicopters land. A haven is a function β that maps each set X of police locations to one of the connected components of the subgraph formed by deleting X; a robber can evade the police by moving in each round of the game to a vertex within this component. Havens must satisfy a consistency property (corresponding to the requirement that the robber cannot move through vertices on which police have already landed): if X is a subset of Y, and both X and Y are valid sets of locations for the given set of police, then β(X) must be a superset of β(Y). A haven has order k if the collection of police locations for which it provides an escape strategy includes all subsets of fewer than k vertices in the graph; in particular, it has order ℵ₀ if it maps every finite subset X of vertices to a component of G \ X. Every ray in G corresponds to a haven of order ℵ₀, namely, the function β that maps every finite set X to the unique component of G \ X that contains infinitely many vertices of the ray. Conversely, every haven of order ℵ₀ can be defined in this way by a ray.^[4] Two rays are equivalent if and only if they define the same haven, so the ends of a graph are in one to one correspondence with its havens of order ℵ₀.

Examples

Part of an infinite grid graph, with vertices at the points where two grid lines meet. Despite having many different rays, it has only one end.

If the infinite graph G is itself a ray, then it has infinitely many ray subgraphs, one starting from each vertex of G. However, all of these rays are equivalent to each other, so G only has one end.

If G is a forest (that is, a graph with no finite cycles), then the intersection of any two rays is either a path or a ray; two rays are equivalent if their intersection is a ray. If a base vertex is chosen in each connected component of G, then each end of G contains a unique ray starting from one of the base vertices, so the ends may be placed in one-to-one correspondence with these canonical rays. Every countable graph G has a spanning forest with the same set of ends as G.^[5] However, there exist uncountably infinite graphs with only one end in which every spanning tree has infinitely many ends.^[6]

If G is an infinite grid graph, then it has many rays, and arbitrarily large sets of vertex-disjoint rays. However, it has only one end. This may be seen most easily using the characterization of ends in terms of havens: the removal of any finite set of vertices leaves exactly one infinite connected component, so there is only one haven (the one that maps each finite set to the unique infinite connected component).

Relation to topological ends

In point-set topology, there is a concept of an end that is similar to, but not quite the same as, the concept of an end in graph theory, dating back much earlier to Template:Harvtxt. If a topological space can be covered by a nested sequence of compact sets $κ_{0} \subset κ_{1} \subset κ_{2} \dots$ , then an end of the space is a sequence of components $U_{0} \supset U_{1} \supset U_{2} \dots$ of the complements of the compact sets. This definition does not depend on the choice of the compact sets: the ends defined by one such choice may be placed in one-to-one correspondence with the ends defined by any other choice.

An infinite graph G may be made into a topological space in two different but related ways:

Replacing each vertex of the graph by a point and each edge of the graph by an open unit interval produces a Hausdorff space from the graph in which a set S is defined to be open whenever each intersection of S with an edge of the graph is an open subset of the unit interval.
Replacing each vertex of the graph by a point and each edge of the graph by a point produces a non-Hausdorff space in which the open sets are the sets S with the property that, if a vertex v of G belongs to S, then so does every edge having v as one of its endpoints.

In either case, every finite subgraph of G corresponds to a compact subspace of the topological space, and every compact subspace corresponds to a finite subgraph together with, in the Hausdorff case, finitely many compact proper subsets of edges. Thus, a graph may be covered by a nested sequence of compact sets if and only if it is locally finite, having a finite number of edges at every vertex.

If a graph G is connected and locally finite, then it has a compact cover in which the set κ_i is the set of vertices at distance at most i from some arbitrarily chosen starting vertex. In this case any haven β defines an end of the topological space in which $U_{i} = β (κ_{i})$ . And conversely, if $U_{0} \supset U_{1} \supset U_{2} \dots$ is an end of the topological space defined from G, it defines a haven in which β(X) is the component containing U_i, where i is any number large enough that κ_i contains X. Thus, for connected and locally finite graphs, the topological ends are in one-to-one correspondence with the graph-theoretic ends.^[7]

For graphs that may not be locally finite, it is still possible to define a topological space from the graph and its ends. This space can be represented as a metric space if and only if the graph has a normal spanning tree, a rooted spanning tree such that each graph edge connects an ancestor-descendant pair. If a normal spanning tree exists, it has the same set of ends as the given graph: each end of the graph must contain exactly one infinite path in the tree.Template:Sfnp

Free ends

An end E of a graph G is defined to be a free end if there is a finite set X of vertices with the property that X separates E from all other ends of the graph. (That is, in terms of havens, β_E(X) is disjoint from β_D(X) for every other end D.) In a graph with finitely many ends, every end must be free. Template:Harvtxt proves that, if G has infinitely many ends, then either there exists an end that is not free, or there exists an infinite family of rays that share a common starting vertex and are otherwise disjoint from each other.

Thick ends

A thick end of a graph G is an end that contains infinitely many pairwise-disjoint rays. Halin's grid theorem characterizes the graphs that contain thick ends: they are exactly the graphs that have a subdivision of the hexagonal tiling as a subgraph.^[8]

Symmetric and almost-symmetric graphs

Template:Harvtxt defines a connected locally-finite graph to be "almost symmetric" if there exist a vertex v and a number D such that, for every other vertex w, there is an automorphism of the graph for which the image of v is within distance D of w; equivalently, a connected locally-finite graph is almost symmetric if its automorphism group has finitely many orbits. As he shows, for every connected locally-finite almost-symmetric graph, the number of ends is either at most two or uncountable; if it is uncountable, the ends have the topology of a Cantor set. Additionally, Mohar shows that the number of ends controls the Cheeger constant

h = inf {\frac{| \partial V |}{| V |}},

where V ranges over all finite nonempty sets of vertices of the graph and where $\partial V$ denotes the set of edges with one endpoint in V. For almost-symmetric graphs with uncountably many ends, h > 0; however, for almost-symmetric graphs with only two ends, h = 0.

Ends of groups

The Cayley graph of the free group on two generators a and b. The ends of the group are in one-to-one correspondence with the rays (infinite paths) from the identity element e to the fringes of the drawing.

Every group and a set of generators for the group determine a Cayley graph, a graph whose vertices are the group elements and the edges are pairs of elements (x,gx) where g is one of the generators. In the case of a finitely-generated group, the ends of the group are defined to be the ends of the Cayley graph for the finite set of generators; this definition is invariant under the choice of generators, in the sense that if two different finite set of generators are chosen, the ends of the two Cayley graphs are in one-to-one correspondence with each other.

For instance, every free group has a Cayley graph (for its free generators) that is a tree. The free group on one generator has a doubly-infinite path as its Cayley graph, with two ends. Every other free group has infinitely many ends.

Every finitely-generated infinite group has either 1, 2, or infinitely many ends, and the Stallings theorem about ends of groups provides a decomposition of groups with more than one end.^[9] In particular:

A finitely-generated infinite group has 2 ends if and only if it has a cyclic subgroup of finite index.
A finitely-generated infinite group has infinitely many ends if and only if it is either a nontrivial free product with amalgamation or HNN-extension with finite amalgamation.
All other finitely-generated infinite groups have exactly one end.

Notes

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References

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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↑ However, as Template:Harvtxt point out, ends of graphs were already considered by Template:Harvtxt.
↑ E.g., this is the form of the equivalence relation used by Template:Harvtxt.
↑ The haven nomenclature, and the fact that two rays define the same haven if and only if they are equivalent, is due to Template:Harvtxt. Template:Harvtxt proved that every haven comes from an end, completing the bijection between ends and havens, using a different nomenclature in which they called havens "directions".
↑ The proof by Template:Harvtxt that every haven can be defined by a ray is nontrivial and involves two cases. If the set $S = ⋂_{X} (β (X) \cup X)$ (where X ranges over all finite sets of vertices) is infinite, then there exists a ray that passes through infinitely many vertices of S, which necessarily determines β. On the other hand, if S is finite, then Template:Harvtxt show that in this case there exists a sequence of finite sets X_i that separate the end from all points whose distance from an arbitrarily chosen starting point in G \ S is i. In this case, the haven is defined by any ray that is followed by a robber using the haven to escape police who land at set X_i in round i of the pursuit-evasion game.
↑ More precisely, in the original formulation of this result by Template:Harvtxt in which ends are defined as equivalence classes of rays, every equivalence class of rays of G contains a unique nonempty equivalence class of rays of the spanning forest. In terms of havens, there is a one-to-one correspondence of havens of order ℵ₀ between G and its spanning tree T for which $β_{T} (X) \subset β_{G} (X)$ for every finite set X and every corresponding pair of havens β_T and β_G.
↑ Template:Harvtxt; Template:Harvtxt; Template:Harvtxt.
↑ Template:Harvtxt.
↑ Template:Harvtxt; Template:Harvtxt.
↑ Template:Harvs.

[1] However, as Template:Harvtxt point out, ends of graphs were already considered by Template:Harvtxt.

[2] E.g., this is the form of the equivalence relation used by Template:Harvtxt.

[3] The haven nomenclature, and the fact that two rays define the same haven if and only if they are equivalent, is due to Template:Harvtxt. Template:Harvtxt proved that every haven comes from an end, completing the bijection between ends and havens, using a different nomenclature in which they called havens "directions".

[4] The proof by Template:Harvtxt that every haven can be defined by a ray is nontrivial and involves two cases. If the set $S = ⋂_{X} (β (X) \cup X)$ (where X ranges over all finite sets of vertices) is infinite, then there exists a ray that passes through infinitely many vertices of S, which necessarily determines β. On the other hand, if S is finite, then Template:Harvtxt show that in this case there exists a sequence of finite sets X_i that separate the end from all points whose distance from an arbitrarily chosen starting point in G \ S is i. In this case, the haven is defined by any ray that is followed by a robber using the haven to escape police who land at set X_i in round i of the pursuit-evasion game.

[5] More precisely, in the original formulation of this result by Template:Harvtxt in which ends are defined as equivalence classes of rays, every equivalence class of rays of G contains a unique nonempty equivalence class of rays of the spanning forest. In terms of havens, there is a one-to-one correspondence of havens of order ℵ₀ between G and its spanning tree T for which $β_{T} (X) \subset β_{G} (X)$ for every finite set X and every corresponding pair of havens β_T and β_G.

[6] Template:Harvtxt; Template:Harvtxt; Template:Harvtxt.

[7] Template:Harvtxt.

[8] Template:Harvtxt; Template:Harvtxt.

[9] Template:Harvs.

[1]

[2]

[3]

[4]

[5]

[6]

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Weyl law: Difference between revisions

en>Myasuda

Revision as of 23:13, 15 December 2013

Contents

Definition and characterization

Examples

Relation to topological ends

Free ends

Thick ends

Symmetric and almost-symmetric graphs

Ends of groups

Notes

References

Navigation menu

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+In the [[mathematics]] of [[infinite graph]]s, an '''end''' of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as [[equivalence class]]es of infinite [[path (graph theory)|paths]], as [[Haven (graph theory)|haven]]s describing strategies for [[pursuit-evasion]] games on the graph, or (in the case of locally finite graphs) as [[end (topology)|topological end]]s of [[topological space]]s associated with the graph.
+Ends of graphs may be used (via [[Cayley graph]]s) to define ends of [[finitely generated group]]s. Finitely generated infinite groups have one, two, or infinitely many ends, and the [[Stallings theorem about ends of groups]] provides a decomposition for groups with more than one end.
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+==Definition and characterization==
+Ends of graphs were defined by {{harvs|first=Rudolf|last=Halin|authorlink=Rudolf Halin|year=1964|txt}} in terms of equivalence classes of infinite paths.<ref>However, as {{harvtxt|Krön|Möller|2008}} point out, ends of graphs were already considered by {{harvtxt|Freudenthal|1945}}.</ref> A '''{{visible anchor|ray}}''' in an infinite graph is a semi-infinite [[simple path (graph theory)|simple path]]; that is, it is an infinite sequence of vertices ''v''<sub>0</sub>, ''v''<sub>1</sub>, ''v''<sub>2</sub>, ... in which each vertex appears at most once in the sequence and each two consecutive vertices in the sequence are the two endpoints of an edge in the graph. According to Halin's definition, two rays ''r''<sub>0</sub> and ''r''<sub>1</sub> are equivalent if there is another ray ''r''<sub>2</sub> (not necessarily different from either of the first two rays) that contains infinitely many of the vertices in each of ''r''<sub>0</sub> and ''r''<sub>1</sub>. This is an [[equivalence relation]]: each ray is equivalent to itself, the definition is symmetric with regard to the ordering of the two rays, and it can be shown to be [[transitive relation|transitive]]. Therefore, it partitions the set of all rays into [[equivalence classes]], and Halin defined an end as one of these equivalence classes.
+An alternative definition of the same equivalence relation has also been used:<ref>E.g., this is the form of the equivalence relation used by {{harvtxt|Diestel|Kühn|2003}}.</ref> two rays ''r''<sub>0</sub> and ''r''<sub>1</sub> are equivalent if there is no finite set ''X'' of vertices that [[Vertex separator|separates]] infinitely many vertices of ''r''<sub>0</sub> from infinitely many vertices of ''r''<sub>1</sub>. This is equivalent to Halin's definition: if the ray ''r''<sub>2</sub> from Halin's definition exists, then any separator must contain infinitely many points of ''r''<sub>2</sub> and therefore cannot be finite, and conversely if ''r''<sub>2</sub> does not exist then a path that alternates as many times as possible between ''r''<sub>0</sub> and ''r''<sub>1</sub> must form the desired finite separator.
+Ends also have a more concrete characterization in terms of [[Haven (graph theory)|havens]], functions that describe evasion strategies for [[pursuit-evasion]] games on a graph ''G''.<ref>The haven nomenclature, and the fact that two rays define the same haven if and only if they are equivalent, is due to {{harvtxt|Robertson|Seymour|Thomas|1991}}. {{harvtxt|Diestel|Kühn|2003}} proved that every haven comes from an end, completing the bijection between ends and havens, using a different nomenclature in which they called havens "directions".</ref>  In the game in question, a robber is trying to evade a set of policemen by moving from vertex to vertex along the edges of ''G''. The police have helicopters and therefore do not need to follow the edges; however the robber can see the police coming and can choose where to move next before the helicopters land. A haven is a function &beta; that maps each set ''X'' of police locations to one of the connected components of the subgraph formed by deleting ''X''; a robber can evade the police by moving in each round of the game to a vertex within this component. Havens must satisfy a consistency property (corresponding to the requirement that the robber cannot move through vertices on which police have already landed): if ''X'' is a subset of ''Y'', and both ''X'' and ''Y'' are valid sets of locations for the given set of police, then &beta;(''X'') must be a superset of &beta;(''Y''). A haven has order ''k'' if the collection of police locations for which it provides an escape strategy includes all subsets of fewer than ''k'' vertices in the graph; in particular, it has order [[Aleph number|ℵ<sub>0</sub>]] if it maps every finite subset ''X'' of vertices to a component of ''G''&nbsp;\&nbsp;''X''. Every ray in ''G'' corresponds to a haven of order ℵ<sub>0</sub>, namely, the function &beta; that maps every finite set ''X'' to the unique component of ''G''&nbsp;\&nbsp;''X'' that contains infinitely many vertices of the ray. Conversely, every haven of order ℵ<sub>0</sub> can be defined in this way by a ray.<ref>The proof by {{harvtxt|Diestel|Kühn|2003}} that every haven can be defined by a ray is nontrivial and involves two cases. If the set <math>S=\bigcap_X\left(\beta(X)\cup X\right)</math> (where ''X'' ranges over all finite sets of vertices) is infinite, then there exists a ray that passes through infinitely many vertices of ''S'', which necessarily determines &beta;. On the other hand, if ''S'' is finite, then {{harvtxt|Diestel|Kühn|2003}} show that in this case there exists a sequence of finite sets ''X''<sub>''i''</sub> that separate the end from all points whose distance from an arbitrarily chosen starting point in ''G''&nbsp;\&nbsp;''S'' is ''i''. In this case, the haven is defined by any ray that is followed by a robber using the haven to escape police who land at set ''X''<sub>''i''</sub> in round ''i'' of the pursuit-evasion game.</ref> Two rays are equivalent if and only if they define the same haven, so the ends of a graph are in one to one correspondence with its havens of order ℵ<sub>0</sub>.
+==Examples==
+[[File:Typy kultury organizacyjnej, Ateny II (ubt).svg|thumb|Part of an infinite [[grid graph]], with vertices at the points where two grid lines meet. Despite having many different rays, it has only one end.]]
+If the infinite graph ''G'' is itself a ray, then it has infinitely many ray subgraphs, one starting from each vertex of ''G''. However, all of these rays are equivalent to each other, so ''G'' only has one end.
+If ''G'' is a forest (that is, a graph with no finite cycles), then the intersection of any two rays is either a path or a ray; two rays are equivalent if their intersection is a ray. If a base vertex is chosen in each connected component of ''G'', then each end of ''G'' contains a unique ray starting from one of the base vertices, so the ends may be placed in one-to-one correspondence with these canonical rays. Every countable graph ''G'' has a [[spanning forest]] with the same set of ends as ''G''.<ref>More precisely, in the original formulation of this result by {{harvtxt|Halin|1964}} in which ends are defined as equivalence classes of rays, every equivalence class of rays of ''G'' contains a unique nonempty equivalence class of rays of the spanning forest. In terms of havens, there is a one-to-one correspondence of havens of order ℵ<sub>0</sub> between ''G'' and its spanning tree ''T'' for which <math>\beta_T(X)\subset \beta_G(X)</math> for every finite set ''X'' and every corresponding pair of havens &beta;<sub>''T''</sub> and &beta;<sub>''G''</sub>.</ref> However, there exist uncountably infinite graphs with only one end in which every spanning tree has infinitely many ends.<ref>{{harvtxt|Seymour|Thomas|1991}}; {{harvtxt|Thomassen|1992}}; {{harvtxt|Diestel|1992}}.</ref>
+If ''G'' is an infinite [[grid graph]], then it has many rays, and arbitrarily large sets of vertex-disjoint rays. However, it has only one end. This may be seen most easily using the characterization of ends in terms of havens: the removal of any finite set of vertices leaves exactly one infinite connected component, so there is only one haven (the one that maps each finite set to the unique infinite connected component).
+==Relation to topological ends==
+In [[point-set topology]], there is a concept of an end that is similar to, but not quite the same as, the concept of an end in graph theory, dating back much earlier to {{harvtxt|Freudenthal|1931}}. If a topological space can be covered by a nested sequence of [[compact set]]s <math>\kappa_0\subset\kappa_1\subset\kappa_2\dots</math>, then an end of the space is a sequence of components <math>U_0\supset U_1\supset U_2\dots</math> of the complements of the compact sets. This definition does not depend on the choice of the compact sets: the ends defined by one such choice may be placed in one-to-one correspondence with the ends defined by any other choice.
+An infinite graph ''G'' may be made into a topological space in two different but related ways:
+*Replacing each vertex of the graph by a point and each edge of the graph by an open [[unit interval]] produces a [[Hausdorff space]] from the graph in which a set ''S'' is defined to be open whenever each intersection of ''S'' with an edge of the graph is an open subset of the unit interval.
+*Replacing each vertex of the graph by a point and each edge of the graph by a point produces a non-Hausdorff space in which the open sets are the sets ''S'' with the property that, if a vertex ''v'' of ''G'' belongs to ''S'', then so does every edge having ''v'' as one of its endpoints.
+In either case, every finite subgraph of ''G'' corresponds to a compact subspace of the topological space, and every compact subspace corresponds to a finite subgraph together with, in the Hausdorff case, finitely many compact proper subsets of edges. Thus, a graph may be covered by a nested sequence of compact sets if and only if it is locally finite, having a finite number of edges at every vertex.
+If a graph ''G'' is connected and locally finite, then it has a compact cover in which the set &kappa;<sub>''i''</sub> is the set of vertices at distance at most ''i'' from some arbitrarily chosen starting vertex. In this case any haven &beta; defines an end of the topological space in which <math>U_i=\beta(\kappa_i)</math>. And conversely, if <math>U_0\supset U_1\supset U_2\dots</math> is an end of the topological space defined from ''G'', it defines a haven in which &beta;(''X'') is the component containing ''U''<sub>''i''</sub>, where ''i'' is any number large enough that &kappa;<sub>''i''</sub> contains ''X''. Thus, for connected and locally finite graphs, the topological ends are in one-to-one correspondence with the graph-theoretic ends.<ref>{{harvtxt|Diestel|Kühn|2003}}.</ref>
+For graphs that may not be locally finite, it is still possible to define a topological space from the graph and its ends. This space can be represented as a [[metric space]] if and only if the graph has a [[Trémaux tree|normal spanning tree]], a rooted [[spanning tree]] such that each graph edge connects an ancestor-descendant pair. If a normal spanning tree exists, it has the same set of ends as the given graph: each end of the graph must contain exactly one infinite path in the tree.{{sfnp|Diestel|2006}}
+==Free ends==
+An end ''E'' of a graph ''G'' is defined to be a '''free end''' if there is a finite set ''X'' of vertices with the property that ''X'' separates ''E'' from all other ends of the graph. (That is, in terms of havens, &beta;<sub>''E''</sub>(''X'') is disjoint from &beta;<sub>''D''</sub>(''X'') for every other end ''D''.) In a graph with finitely many ends, every end must be free. {{harvtxt|Halin|1964}} proves that, if ''G'' has infinitely many ends, then either there exists an end that is not free, or there exists an infinite family of rays that share a common starting vertex and are otherwise disjoint from each other.
+==Thick ends==
+A thick end of a graph ''G'' is an end that contains infinitely many pairwise-[[disjoint sets|disjoint]] rays. [[Halin's grid theorem]] characterizes the graphs that contain thick ends: they are exactly the graphs that have a [[Homeomorphism (graph theory)|subdivision]] of the [[hexagonal tiling]] as a subgraph.<ref>{{harvtxt|Halin|1965}}; {{harvtxt|Diestel|2004}}.</ref>
+==Symmetric and almost-symmetric graphs==
+{{harvtxt|Mohar|1991}} defines a connected locally-finite graph to be "almost symmetric" if there exist a vertex ''v'' and a number ''D'' such that, for every other vertex ''w'', there is an [[graph isomorphism|automorphism]] of the graph for which the image of ''v'' is within distance ''D'' of ''w''; equivalently, a connected locally-finite graph is almost symmetric if its automorphism group has finitely many orbits. As he shows, for every connected locally-finite almost-symmetric graph, the number of ends is either at most two or uncountable; if it is uncountable, the ends have the topology of a [[Cantor set]]. Additionally, Mohar shows that the number of ends controls the [[Cheeger constant (graph theory)|Cheeger constant]]
+:<math>h=\inf\left\{\frac{|\partial V|}{|V|}\right\},</math>
+where ''V'' ranges over all finite nonempty sets of vertices of the graph and
+where <math>\partial V</math> denotes the set of edges with one endpoint in ''V''. For almost-symmetric graphs with uncountably many ends, ''h''&nbsp;>&nbsp;0; however, for almost-symmetric graphs with only two ends, ''h''&nbsp;=&nbsp;0.
+==Ends of groups==
+[[Image:Cayley graph of F2.svg|right|thumb|The Cayley graph of the [[free group]] on two generators ''a'' and ''b''. The ends of the group are in one-to-one correspondence with the rays (infinite paths) from the identity element ''e'' to the fringes of the drawing.]]
+Every [[group (mathematics)|group]] and a set of generators for the group determine a [[Cayley graph]], a graph whose vertices are the group elements and the edges are pairs of elements (''x'',''gx'') where ''g'' is one of the generators. In the case of a [[finitely-generated group]], the ends of the group are defined to be the ends of the Cayley graph for the finite set of generators; this definition is invariant under the choice of generators, in the sense that if two different finite set of generators are chosen, the ends of the two Cayley graphs are in one-to-one correspondence with each other.
+For instance, every [[free group]] has a Cayley graph (for its free generators) that is a tree. The free group on one generator has a doubly-infinite path as its Cayley graph, with two ends. Every other free group has infinitely many ends.
+Every finitely-generated infinite group has either 1, 2, or infinitely many ends, and the [[Stallings theorem about ends of groups]] provides a decomposition of groups with more than one end.<ref>{{harvs|last=Stallings|year=1968|year2=1971|txt}}.</ref>  In particular:
+# A finitely-generated infinite group has 2 ends if and only if it has a [[cyclic group|cyclic]] [[subgroup]] of finite [[index of a subgroup|index]].
+# A finitely-generated infinite group has infinitely many ends if and only if it is either a nontrivial [[free product with amalgamation]] or [[HNN-extension]] with finite amalgamation.
+# All other finitely-generated infinite groups have exactly one end.
+==Notes==
+{{reflist}}
+==References==
+*{{citation
+ | last = Diestel | first = Reinhard
+ | doi = 10.1016/0012-365X(92)90650-5
+ | issue = 1-3
+ | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
+ | mr = 1172358
+ | pages = 313–327
+ | title = The end structure of a graph: recent results and open problems
+ | volume = 100
+ | year = 1992}}.
+*{{citation
+ | last = Diestel | first = Reinhard
+ | doi = 10.1007/BF02941538
+ | journal = Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
+ | mr = 2112834
+ | pages = 237–242
+ | title = A short proof of Halin's grid theorem
+ | volume = 74
+ | year = 2004}}.
+*{{citation
+ | last = Diestel | first = Reinhard
+ | doi = 10.1016/j.jctb.2006.02.010
+ | issue = 6
+ | journal = [[Journal of Combinatorial Theory]]
+ | mr = 2274079
+ | pages = 846–854
+ | series = Series B
+ | title = End spaces and spanning trees
+ | volume = 96
+ | year = 2006}}.
+*{{citation
+ | last1 = Diestel | first1 = Reinhard
+ | last2 = Kühn | first2 = Daniela | author2-link = Daniela Kühn
+ | doi = 10.1016/S0095-8956(02)00034-5
+ | issue = 1
+ | journal = [[Journal of Combinatorial Theory]] | series = Series B
+ | mr = 1967888
+ | pages = 197–206
+ | title = Graph-theoretical versus topological ends of graphs
+ | volume = 87
+ | year = 2003}}.
+*{{citation
+ | last = Freudenthal | first = Hans | authorlink = Hans Freudenthal
+ | journal = [[Mathematische Zeitschrift]]
+ | pages = 692–713
+ | title = Über die Enden topologischer Räume und Gruppen
+ | volume = 33
+ | year = 1931
+ | doi=10.1007/BF01174375}}.
+*{{citation
+ | last = Freudenthal | first = Hans | authorlink = Hans Freudenthal
+ | journal = Commentarii Mathematici Helvetici
+ | mr = 0012214
+ | pages = 1–38
+ | title = Über die Enden diskreter Räume und Gruppen
+ | volume = 17
+ | year = 1945}}.
+*{{citation
+ | last = Halin | first = Rudolf | authorlink = Rudolf Halin
+ | journal = [[Mathematische Annalen]]
+ | mr = 0170340
+ | pages = 125–137
+ | title = Über unendliche Wege in Graphen
+ | volume = 157
+ | year = 1964}}.
+*{{citation
+ | last = Halin | first = Rudolf | authorlink = Rudolf Halin
+ | doi = 10.1002/mana.19650300106
+ | journal = [[Mathematische Nachrichten]]
+ | mr = 0190031
+ | pages = 63–85
+ | title = Über die Maximalzahl fremder unendlicher Wege in Graphen
+ | volume = 30
+ | year = 1965}}.
+*{{citation
+ | last1 = Krön | first1 = Bernhard
+ | last2 = Möller | first2 = Rögnvaldur G.
+ | doi = 10.1002/mana.200510587
+ | issue = 1
+ | journal = [[Mathematische Nachrichten]]
+ | mr = 2376468
+ | pages = 62–74
+ | title = Metric ends, fibers and automorphisms of graphs
+ | url = http://homepage.univie.ac.at/bernhard.kroen/kroen_metric.pdf
+ | volume = 281
+ | year = 2008}}.
+*{{citation
+ | last = Mohar | first = Bojan | authorlink = Bojan Mohar
+ | doi = 10.1016/0012-365X(91)90337-2
+ | issue = 1-3
+ | journal = Discrete Mathematics
+ | mr = 1141939
+ | pages = 193–219
+ | title = Some relations between analytic and geometric properties of infinite graphs
+ | url = http://www.fmf.uni-lj.si/~mohar/Reprints/1991/BM91_Mohar_DM95_InfiniteGraphs.pdf
+ | volume = 95
+ | year = 1991}}.
+*{{citation
+ | last1 = Robertson | first1 = Neil | author1-link = Neil Robertson (mathematician)
+ | last2 = Seymour | first2 = Paul | author2-link = Paul Seymour (mathematician)
+ | last3 = Thomas | first3 = Robin | author3-link = Robin Thomas (mathematician)
+ | doi = 10.1016/0012-365X(91)90343-Z
+ | issue = 1-3
+ | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
+ | mr = 1141945
+ | pages = 303–319
+ | title = Excluding infinite minors
+ | volume = 95
+ | year = 1991}}.
+*{{citation
+ | last1 = Seymour | first1 = Paul | author1-link = Paul Seymour (mathematician)
+ | last2 = Thomas | first2 = Robin | author2-link = Robin Thomas (mathematician)
+ | doi = 10.2307/2048796
+ | issue = 4
+ | journal = [[Proceedings of the American Mathematical Society]]
+ | mr = 1045600
+ | pages = 1163–1171
+ | title = An end-faithful spanning tree counterexample
+ | volume = 113
+ | year = 1991}}.
+*{{citation
+ | last = Stallings | first = John R. | authorlink = John R. Stallings
+ | journal = [[Annals of Mathematics]] | series = Second Series
+ | mr = 0228573
+ | pages = 312–334
+ | title = On torsion-free groups with infinitely many ends
+ | volume = 88
+ | year = 1968}}.
+*{{citation
+ | last = Stallings | first = John R. | authorlink = John R. Stallings
+ | location = New Haven, Conn.
+ | mr = 0415622
+ | publisher = Yale University Press
+ | series = Yale Mathematical Monographs
+ | title = Group theory and three-dimensional manifolds: A James K. Whittemore Lecture in Mathematics given at Yale University, 1969
+ | volume = 4
+ | year = 1971}}.
+*{{citation
+ | last = Thomassen | first = Carsten | authorlink = Carsten Thomassen
+ | doi = 10.1016/0095-8956(92)90059-7
+ | issue = 2
+ | journal = [[Journal of Combinatorial Theory]] | series = Series B
+ | mr = 1152455
+ | pages = 322–324
+ | title = Infinite connected graphs with no end-preserving spanning trees
+ | volume = 54
+ | year = 1992}}.
+[[Category:Graph theory objects]]
+[[Category:Infinite graphs]]

Weyl law: Difference between revisions

en>Myasuda

Revision as of 23:13, 15 December 2013

Definition and characterization

Examples

Relation to topological ends

Free ends

Thick ends

Symmetric and almost-symmetric graphs

Ends of groups

Notes

References

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