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A highly symmetrical graph, the Petersen graph, which is vertex-transitive, symmetric, distance-transitive, and distance-regular. It has diameter 2. Its automorphism group has 120 elements, and is in fact the symmetric group $S_{5}$ .

Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants.

Branches of algebraic graph theory

Using linear algebra

The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties. As a simple example, a connected graph with diameter D will have at least D+1 distinct values in its spectrum.^[1] Aspects of graph spectra have been used in analysing the synchronizability of networks.

Using group theory

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A Cayley graph for the alternating group A₄, forming a truncated tetrahedron in three dimensions. All Cayley graphs are vertex-transitive, but some vertex-transitive graphs (like the Petersen graph) are not Cayley graphs.

A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible. According to the chromatic polynomial, there are 120 such colorings with 3 colors.

This second branch of algebraic graph theory is related to the first, since the symmetry properties of a graph are reflected in its spectrum. In particular, the spectrum of a highly symmetrical graph, such as the Petersen graph, has few distinct values^[1] (the Petersen graph has 3, which is the minimum possible, given its diameter). For Cayley graphs, the spectrum can be related directly to the structure of the group, in particular to its irreducible characters.^[1]^[3]

Studying graph invariants

Finally, the third branch of algebraic graph theory concerns algebraic properties of invariants of graphs, and especially the chromatic polynomial, the Tutte polynomial and knot invariants. The chromatic polynomial of a graph, for example, counts the number of its proper vertex colorings. For the Petersen graph, this polynomial is $t (t - 1) (t - 2) (t^{7} - 12 t^{6} + 67 t^{5} - 230 t^{4} + 529 t^{3} - 814 t^{2} + 775 t - 352)$ .^[1] In particular, this means that the Petersen graph cannot be properly colored with one or two colors, but can be colored in 120 different ways with 3 colors. Much work in this area of algebraic graph theory was motivated by attempts to prove the four color theorem. However, there are still many open problems, such as characterizing graphs which have the same chromatic polynomial, and determining which polynomials are chromatic.

References

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Template:Refend

↑ ^1.0 ^1.1 ^1.2 ^1.3 ^1.4 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.

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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
↑ R. Frucht. Graphs of Degree 3 with given abstract group, Canad. J. Math. 3 1949.
↑ *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

[biggs-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 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

[2] R. Frucht. Graphs of Degree 3 with given abstract group, Canad. J. Math. 3 1949.

[babai-3] *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

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+[[Image:Petersen1 tiny.svg|thumb|200px|A highly symmetrical graph, the [[Petersen graph]], which is [[vertex-transitive graph|vertex-transitive]], [[symmetric graph|symmetric]], [[distance-transitive graph|distance-transitive]], and [[distance-regular graph|distance-regular]].  It has [[diameter]] 2. Its [[Graph automorphism|automorphism group]] has 120 elements, and is in fact the [[symmetric group]] <math>S_5</math>.]]
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+'''Algebraic [[graph theory]]''' is a branch of [[mathematics]] in which [[algebra]]ic methods are applied to problems about [[Graph (mathematics)|graphs]]. This is in contrast to [[geometric graph theory|geometric]], [[combinatorics#Graph theory|combinatoric]], or [[algorithmic graph theory|algorithmic]] approaches.  There are three main branches of algebraic graph theory, involving the use of [[linear algebra]], the use of [[group theory]], and the study of [[graph property|graph invariants]].
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+==Branches of algebraic graph theory==
+=== Using linear algebra ===
+The first branch of algebraic graph theory involves the study of graphs in connection with [[linear algebra]]. Especially, it studies the [[Eigendecomposition of a matrix|spectrum]] of the [[adjacency matrix]], or the [[Laplacian matrix]] of a graph (this part of algebraic graph theory is also called [[spectral graph theory]]).  For the [[Petersen graph]], for example, the spectrum of the adjacency matrix is (−2,&nbsp;−2,&nbsp;−2,&nbsp;−2,&nbsp;1,&nbsp;1,&nbsp;1,&nbsp;1,&nbsp;1,&nbsp;3). Several theorems relate properties of the spectrum to other [[graph property|graph properties]].  As a simple example, a [[connectivity (graph theory)|connected]] graph with [[diameter]] ''D'' will have at least ''D''+1 distinct values in its spectrum.<ref name="biggs">{{Citation | author=Biggs, Norman | title=Algebraic Graph Theory | edition=2nd | location=Cambridge | publisher=Cambridge University Press | year=1993 | isbn=0-521-45897-8 | postscript=<!--none-->}}</ref> [[Algebraic connectivity|Aspects]] of graph spectra have been used in analysing the [[synchronizability]] of [[Network theory|networks]].
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+===Using group theory===
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+{{Graph families defined by their automorphisms}}
- <ul>
+The second branch of algebraic graph theory involves the study of graphs in connection to [[group theory]], particularly [[Graph automorphism|automorphism groups]] and [[geometric group theory]]. The focus is placed on various families of graphs based on [[symmetry]] (such as [[symmetric graph]]s, [[vertex-transitive graph]]s, [[edge-transitive graph]]s, [[distance-transitive graph]]s, [[distance-regular graph]]s, and [[strongly regular graph]]s), and on the inclusion relationships between these families. Certain of such categories of graphs are sparse enough that [[Foster census|lists]] of graphs can be drawn up. By [[Frucht's theorem]], all [[group (mathematics)|groups]] can be represented as the automorphism group of a connected graph (indeed, of a [[cubic graph]]).<ref>[[R. Frucht]]. Graphs of Degree 3 with given abstract group, Canad. J. Math. 3 1949.</ref> Another connection with group theory is that, given any group, symmetrical graphs known as [[Cayley graph]]s can be generated, and these have properties related to the structure of the group.<ref name="biggs" />
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+[[Image:TruncatedTetrahedron.gif|thumb|left|220px|A [[Cayley graph]] for the [[alternating group]] ''A''<sub>4</sub>, forming a [[truncated tetrahedron]] in three dimensions.  All Cayley graphs are [[vertex-transitive graph|vertex-transitive]], but some vertex-transitive graphs (like the [[Petersen graph]]) are not Cayley graphs.]]
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+[[Image:Petersen graph 3-coloring.svg|thumb|right|200px|A proper vertex coloring of the [[Petersen graph]] with 3 colors, the minimum number possible.  According to the [[chromatic polynomial]], there are 120 such colorings with 3 colors.]]
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+This second branch of algebraic graph theory is related to the first, since the symmetry properties of a graph are reflected in its spectrum. In particular, the spectrum of a highly symmetrical graph, such as the Petersen graph, has few distinct values<ref name="biggs" /> (the Petersen graph has 3, which is the minimum possible, given its diameter). For Cayley graphs, the spectrum can be related directly to the structure of the group, in particular to its [[character theory|irreducible characters]].<ref name="biggs" /><ref name="babai">*{{Citation | first = L | last = Babai | editor-last = Graham | editor-first = R | editor2-last = Groetschel | editor2-first = M | editor3-last = Lovasz | editor3-first = L | title = Handbook of Combinatorics | contribution = Automorphism groups, isomorphism, reconstruction | contribution-url = http://www.cs.uchicago.edu/files/tr_authentic/TR-94-10.ps | year = 1996 | publisher = Elsevier | postscript = <!--none-->}}</ref>
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+===Studying graph invariants===
-相关的主题文章：
+Finally, the third branch of algebraic graph theory concerns algebraic properties of [[graph property|invariants]] of graphs, and especially the [[chromatic polynomial]], the [[Tutte polynomial]] and [[knot invariant]]s. The chromatic polynomial of a graph, for example, counts the number of its proper [[vertex coloring]]s.  For the Petersen graph, this polynomial is <math>t(t-1)(t-2)(t^7-12t^6+67t^5-230t^4+529t^3-814t^2+775t-352)</math>.<ref name="biggs" /> In particular, this means that the Petersen graph cannot be properly colored with one or two colors, but can be colored in 120 different ways with 3 colors.  Much work in this area of algebraic graph theory was motivated by attempts to prove the [[four color theorem]]. However, there are still many [[Graph coloring#Open problems|open problems]], such as characterizing graphs which have the same chromatic polynomial, and determining which polynomials are chromatic.
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+==See also==
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+*[[Spectral graph theory]]
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+*[[Algebraic combinatorics]]
- <ul>
+*[[Algebraic connectivity]]
+*[[Dulmage–Mendelsohn decomposition]]
-   <li>[http://ciarcr.org/spip.php?article310/ http://ciarcr.org/spip.php?article310/]</li>
+*[[Graph property]]
-   <li>[http://www.jankloco.com/activity/p/56050/ http://www.jankloco.com/activity/p/56050/]</li>
+==References==
-   <li>[http://www.sebalo.info/spip/spip.php?article13 http://www.sebalo.info/spip/spip.php?article13]</li>
+{{reflist}}
+{{refbegin}}
-   <li>[http://gouzaimaozai.com/forum.php?mod=viewthread&tid=98613&fromuid=14202 http://gouzaimaozai.com/forum.php?mod=viewthread&tid=98613&fromuid=14202]</li>
+*{{citation|first1=Chris|last1=Godsil|authorlink1=Chris Godsil|first2=Gordon|last2=Royle|authorlink2=Gordon Royle|title=Algebraic Graph Theory|series=Graduate Texts in Mathematics|volume=207|publisher=Springer-Verlag|location=New York|year=2001|postscript=<!--none-->}}.
+{{refend}}
- </ul>
+[[Category:Algebraic graph theory|*]]

Q-analog: Difference between revisions

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Contents

Branches of algebraic graph theory

Using linear algebra

Using group theory

Studying graph invariants

See also

References

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