Hopf algebra: Difference between revisions
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{{about|sets of vertices connected by edges|graphs of mathematical functions|Graph of a function|other uses|Graph (disambiguation)}} | |||
[[File:6n-graf.svg|thumb|250px|A [[graph drawing|drawing]] of a [[labeled graph]] on 6 vertices and 7 edges.]] | |||
In [[mathematics]], and more specifically in [[graph theory]], a '''graph''' is a representation of a set of objects where some pairs of objects are connected by [[Link (geometry)|links]]. The interconnected objects are represented by mathematical abstractions called ''[[Vertex (graph theory)|vertices]]'', and the links that connect some pairs of vertices are called ''edges''.<ref>{{cite book|last=Trudeau|first=Richard J.|title=Introduction to Graph Theory|year=1993|publisher=Dover Pub.|location=New York|isbn=978-0-486-67870-2|pages=19|url=http://store.doverpublications.com/0486678709.html|edition=Corrected, enlarged republication.|accessdate=8 August 2012|quote=A graph is an object consisting of two sets called its ''vertex set'' and its ''edge set''.}}</ref> Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in [[discrete mathematics]]. | |||
The person | The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this is an undirected graph, because if person A shook hands with person B, then person B also shook hands with person A. In contrast, if there is an edge from person A to person B when person A knows of person B, then this graph is directed, because knowledge of someone is not necessarily a [[symmetric relation]] (that is, one person knowing another person does not necessarily imply the reverse; for example, many fans may know of a celebrity, but the celebrity is unlikely to know of all their fans). This latter type of graph is called a ''directed'' graph and the edges are called ''directed edges'' or ''arcs''. | ||
Vertices are also called ''nodes'' or ''points'', and edges are also called ''arcs'' or ''lines''. Graphs are the basic subject studied by [[graph theory]]. The word "graph" was first used in this sense by [[James Joseph Sylvester|J.J. Sylvester]] in 1878.<ref>See: | |||
* J. J. Sylvester (February 7, 1878) [http://books.google.com/books?id=KcoKAAAAYAAJ&vq=Sylvester&pg=PA284#v=onepage&q&f=false "Chemistry and algebra,"] ''Nature'', '''17''' : 284. From page 284: "Every invariant and covariant thus becomes expressible by a ''graph'' precisely identical with a Kekuléan diagram or chemicograph." | |||
* J. J. Sylvester (1878) [http://books.google.com/books?id=1q0EAAAAYAAJ&pg=PA64#v=onepage&q&f=false "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, — with three appendices,"] ''American Journal of Mathematics, Pure and Applied'', '''1''' (1) : 64-90. The term "graph" first appears in this paper on page 65.</ref><ref>{{Cite book | |||
| title = Handbook of graph theory | |||
| first1 = Jonathan L. | |||
| last1 = Gross | |||
| first2 = Jay | |||
| last2 = Yellen | |||
| publisher = [[CRC Press]] | |||
| year = 2004 | |||
| page = [http://books.google.com/books?id=mKkIGIea_BkC&pg=PA35&lpg=PA35 35] | |||
| isbn = 978-1-58488-090-5 | |||
| url = http://books.google.com/?id=mKkIGIea_BkC | |||
| postscript = <!--None-->}}</ref> | |||
==Definitions== | |||
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. | |||
===Graph=== | |||
In the most common sense of the term,<ref>See, for instance, Iyanaga and Kawada, '''69 J''', p. 234 or Biggs, p. 4.</ref> a '''graph''' is an [[ordered pair]] ''G'' = (''V'', ''E'') comprising a [[set (mathematics)|set]] ''V'' of '''vertices''' or '''nodes''' together with a set ''E'' of '''edges''' or '''lines''', which are 2-element subsets of ''V'' (i.e., an edge is related with two vertices, and the relation is represented as an [[unordered pair]] of the vertices with respect to the particular edge). To avoid ambiguity, this type of graph may be described precisely as [[graph (mathematics)#Undirected graph|undirected]] and [[graph (mathematics)#Simple graph|simple]]'''. | |||
Other senses of ''graph'' stem from different conceptions of the edge set. In one more generalized notion,<ref>See, for instance, Graham et al., p. 5.</ref> ''E'' is a set together with a relation of '''incidence''' that associates with each edge two vertices. In another generalized notion, ''E'' is a [[multiset]] of unordered pairs of (not necessarily distinct) vertices. Many authors call this type of object a [[multigraph]] or pseudograph. | |||
All of these variants and others are described more fully below. | |||
The vertices belonging to an edge are called the '''ends''', '''endpoints''', or '''end vertices''' of the edge. A vertex may exist in a graph and not belong to an edge. | |||
''V'' and ''E'' are usually taken to be finite, and many of the well-known results are not true (or are rather different) for '''infinite graphs''' because many of the arguments fail in the [[infinite graph|infinite case]]. The '''order''' of a graph is <math>|V|</math> (the number of vertices). A graph's '''size''' is <math>|E|</math>, the number of edges. The '''degree''' of a vertex is the number of edges that connect to it, where an edge that connects to the vertex at both ends (a [[loop (graph theory)|loop]]) is counted twice. | |||
For an edge {''u'', ''v''}, graph theorists usually use the somewhat shorter notation ''uv''. | |||
{{-}} | |||
===Adjacency relation=== | |||
The edges ''E'' of an undirected graph ''G'' induce a symmetric binary relation ~ on ''V'' that is called the '''adjacency''' relation of ''G''. Specifically, for each edge {''u'', ''v''} the vertices ''u'' and ''v'' are said to be '''adjacent''' to one another, which is denoted ''u'' ~ ''v''. | |||
==Types of graphs== | |||
===Distinction in terms of the main definition=== | |||
As stated above, in different contexts it may be useful to define the term ''graph'' with different degrees of generality. Whenever it is necessary to draw a strict distinction, the following terms are used. Most commonly, in modern texts in graph theory, unless stated otherwise, ''graph'' means "undirected simple finite graph" (see the definitions below). | |||
{{multiple image | |||
| right | |||
| footer = | |||
| width1 = 125 | |||
| image1 = Directed.svg | |||
| caption1 = A directed graph. | |||
| width2 = 125 | |||
| image2 = Undirected.svg | |||
| caption2 = A [[Graph (mathematics)#Simple graph|simple]] undirected graph with three vertices and three edges. Each vertex has degree two, so this is also a regular graph. | |||
}} | |||
====Undirected graph==== | |||
An undirected graph is one in which edges have no orientation. The edge (a, b) is identical to the edge (b, a), i.e., they are not ordered pairs, but sets {''u'', ''v''} (or 2-multisets) of vertices. The maximum number of edges in an undirected graph without a self-loop is ''n''(''n'' - 1)/2. | |||
====Directed graph==== | |||
{{main|Directed graph}} | |||
A '''directed graph''' or '''digraph''' is an ordered pair ''D'' = (''V'', ''A'') with | |||
* ''V'' a [[set (mathematics)|set]] whose elements are called '''vertices''' or '''nodes''', and | |||
* ''A'' a set of ordered pairs of vertices, called '''arcs''', '''directed edges''', or '''arrows'''. | |||
An arc ''a'' = (''x'', ''y'') is considered to be directed '''from''' ''x'' '''to''' ''y''; ''y'' is called the '''head''' and ''x'' is called the '''tail''' of the arc; ''y'' is said to be a '''direct successor''' of ''x'', and ''x'' is said to be a '''direct predecessor''' of ''y''. If a [[path (graph theory)|path]] leads from ''x'' to ''y'', then ''y'' is said to be a '''successor''' of ''x'' and '''reachable''' from ''x'', and ''x'' is said to be a '''predecessor''' of ''y''. The arc (''y'', ''x'') is called the arc (''x'', ''y'') '''inverted'''. | |||
A directed graph ''D'' is called '''symmetric''' if, for every arc in ''D'', the corresponding inverted arc also belongs to ''D''. A symmetric loopless directed graph ''D'' = (''V'', ''A'') is equivalent to a simple undirected graph ''G'' = (''V'', ''E''), where the pairs of inverse arcs in ''A'' correspond 1-to-1 with the edges in ''E''; thus the edges in ''G'' number |''E''| = |''A''|/2, or half the number of arcs in ''D''. | |||
A variation on this definition is the '''oriented graph''', in which at most one of (''x'', ''y'') and (''y'', ''x'') may be arcs. | |||
====Mixed graph==== | |||
{{main|Mixed graph}} | |||
A '''mixed graph''' ''G'' is a graph in which some edges may be directed and some may be undirected. | |||
It is written as an ordered triple ''G'' = (''V'', ''E'', ''A'') with ''V'', ''E'', and ''A'' defined as above. | |||
Directed and undirected graphs are special cases. | |||
====Multigraph==== | |||
A [[loop (graph theory)|loop]] is an edge (directed or undirected) which starts and ends on the same vertex; these may be permitted or not permitted according to the application. In this context, an edge with two different ends is called a '''link'''. | |||
The term "[[multigraph]]" is generally understood to mean that [[multiple edges]] (and sometimes loops) are allowed. Where graphs are defined so as to ''allow'' loops and multiple edges, a multigraph is often defined to mean a graph ''without'' loops,<ref>For example, see Balakrishnan, p. 1, Gross (2003), p. 4, and Zwillinger, p. 220.</ref> however, where graphs are defined so as to ''disallow'' loops and multiple edges, the term is often defined to mean a "graph" which can have both multiple edges ''and'' loops,<ref>For example, see. Bollobás, p. 7 and Diestel, p. 25.</ref> although many use the term "[[pseudograph]]" for this meaning.<ref>Gross (1998), p. 3, Gross (2003), p. 205, Harary, p.10, and Zwillinger, p. 220.</ref> | |||
====Quiver==== | |||
A '''[[Quiver (mathematics)|quiver]]''' or "multidigraph" is a directed graph which may have more than one arrow from a given source to a given target. A quiver may also have directed loops in it. | |||
====Simple graph==== | |||
As opposed to a multigraph, a simple graph is an undirected graph that has no [[Loop (graph theory)|loops]] (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices. In a simple graph the edges of the graph form a set (rather than a [[multiset]]) and each edge is a ''distinct'' pair of vertices. In a simple graph with ''n'' vertices every vertex has a degree that is less than ''n'' (the converse, however, is not true — there exist non-simple graphs with ''n'' vertices in which every vertex has a degree smaller than ''n''). | |||
====Weighted graph==== | |||
A graph is a [[weighted graph]] if a number (weight) is assigned to each edge.<ref>{{cite book|last1=Fletcher|first1=Peter|last2=Hoyle|first2=Hughes|last3=Patty|first3=C. Wayne|title=Foundations of Discrete Mathematics|year=1991|publisher=PWS-KENT Pub. Co.| location=Boston| isbn=0-53492-373-9| pages=463 | edition=International student ed.|quote=A '''weighted graph''' is a graph in which a number ''w(e)'', called its '''weight''', is assigned to each edge ''e''.}}</ref> Such weights might represent, for example, costs, lengths or capacities, etc. depending on the problem at hand. Some authors call such a graph a network.<ref>{{Citation | last=Strang | first=Gilbert | title=Linear Algebra and Its Applications | publisher=Brooks Cole | edition=4th | year=2005 | isbn=0-03-010567-6 | url=http://books.google.com/books?vid=ISBN0030105676 }}</ref> | |||
[[Weighted_correlation_network_analysis| Weighted correlation networks ]] can be defined by soft-thresholding the pairwise correlations among variables (e.g. gene measurements). | |||
====Half-edges, loose edges==== | |||
In certain situations it can be helpful to allow edges with only one end, called '''half-edges''', or no ends ('''loose edges'''); see for example [[signed graph]]s and [[biased graph]]s. | |||
===Important graph classes=== | |||
====Regular graph==== | |||
{{main|Regular graph}} | |||
A regular graph is a graph where each vertex has the same number of neighbours, i.e., every vertex has the same degree or valency. A regular graph with vertices of degree ''k'' is called a ''k''‑regular graph or regular graph of degree ''k''. | |||
====Complete graph==== | |||
[[File:Complete graph K5.svg|thumb|125px|A complete graph with 5 vertices. Each vertex has an edge to every other vertex.]] | |||
{{Main|Complete graph}} | |||
Complete graphs have the feature that each pair of vertices has an edge connecting them. | |||
====Finite and infinite graphs==== | |||
A finite graph is a graph ''G'' = (''V'', ''E'') such that ''V'' and ''E'' are [[finite set]]s. An infinite graph is one with an [[Infinite set|infinite]] set of vertices or edges or both. | |||
Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated. | |||
====Graph classes in terms of connectivity==== | |||
{{main|Connectivity (graph theory)}} | |||
In an undirected graph ''G'', two [[vertex (graph theory)|vertices]] ''u'' and ''v'' are called '''connected''' if ''G'' contains a [[Path (graph theory)|path]] from ''u'' to ''v''. Otherwise, they are called '''disconnected'''. A graph is called '''connected''' if every pair of distinct vertices in the graph is connected; otherwise, it is called '''disconnected'''. | |||
A graph is called '''[[K-vertex-connected graph|''k''-vertex-connected]]''' or '''[[k-edge-connected graph|''k''-edge-connected]]''' if no set of ''k-1'' vertices (respectively, edges) exists that, when removed, disconnects the graph. A ''k''-vertex-connected graph is often called simply '''''k''-connected'''. | |||
A [[directed graph]] is called '''weakly connected''' if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is '''strongly connected''' or '''strong''' if it contains a directed path from ''u'' to ''v'' and a directed path from ''v'' to ''u'' for every pair of vertices ''u'', ''v''. | |||
====Category of all graphs==== | |||
The [[category theory|category]] of all graphs is the [[slice category]] <math>\mathbf{Set}\downarrow D</math> where <math>D : \mathbf{Set} \rightarrow \mathbf{Set}</math> is the [[functor]] taking a set <math>s</math> to <math>s \times s</math>. | |||
==Properties of graphs== | |||
{{see also|Glossary of graph theory|Graph property}} | |||
Two edges of a graph are called '''adjacent''' if they share a common vertex. Two arrows of a directed graph are called '''consecutive''' if the head of the first one is at the nock (notch end) of the second one. Similarly, two vertices are called '''adjacent''' if they share a common edge ('''consecutive''' if they are at the notch and at the head of an arrow), in which case the common edge is said to '''join''' the two vertices. An edge and a vertex on that edge are called '''incident'''. | |||
The graph with only one vertex and no edges is called the '''trivial graph'''. A graph with only vertices and no edges is known as an '''edgeless graph'''. The graph with no vertices and no edges is sometimes called the '''[[null graph]]''' or '''empty graph''', but the terminology is not consistent and not all mathematicians allow this object. | |||
In a '''weighted''' graph or digraph, each edge is associated with some value, variously called its ''cost'', ''weight'', ''length'' or other term depending on the application; such graphs arise in many contexts, for example in [[Shortest path problem|optimal routing problem]]s such as the [[traveling salesman problem]]. | |||
Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called '''vertex-labeled'''. However, for many questions it is better to treat vertices as indistinguishable; then the graph may be called '''unlabeled'''. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges). The same remarks apply to edges, so graphs with labeled edges are called '''edge-labeled''' graphs. Graphs with labels attached to edges or vertices are more generally designated as '''labeled'''. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called ''unlabeled''. (Note that in the literature the term ''labeled'' may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.) | |||
==Examples== | |||
[[File:6n-graf.svg|thumb|A graph with six nodes.]] | |||
* The diagram at right is a graphic representation of the following graph: | |||
: ''V'' = {1, 2, 3, 4, 5, 6} | |||
: ''E'' = {{1, 2}, {1, 5}, {2, 3}, {2, 5}, {3, 4}, {4, 5}, {4, 6}}. | |||
* In [[category theory]] a [[category (mathematics)|small category]] can be represented by a directed [[multigraph]] in which the objects of the category represented as vertices and the [[morphism]]s as directed edges. Then, the [[functor]]s between categories induce some, but not necessarily all, of the [[digraph morphism]]s of the graph. | |||
* In [[computer science]], directed graphs are used to represent knowledge (e.g., [[Conceptual graph]]), [[finite state machine]]s, and many other discrete structures. | |||
* A [[binary relation]] ''R'' on a set ''X'' defines a directed graph. An element ''x'' of ''X'' is a direct predecessor of an element ''y'' of ''X'' iff ''xRy''. | |||
==Important graphs== | |||
Basic examples are: | |||
* In a [[complete graph]], each pair of vertices is joined by an edge; that is, the graph contains all possible edges. | |||
* In a [[bipartite graph]], the vertex set can be [[Partition of a set|partitioned]] into two sets, ''W'' and ''X'', so that no two vertices in ''W'' are adjacent and no two vertices in ''X'' are adjacent. Alternatively, it is a graph with a [[chromatic number]] of 2. | |||
* In a [[complete bipartite graph]], the vertex set is the union of two disjoint sets, ''W'' and ''X'', so that every vertex in ''W'' is adjacent to every vertex in ''X'' but there are no edges within ''W'' or ''X''. | |||
* In a ''linear graph'' or [[path graph]] of length ''n'', the vertices can be listed in order, ''v''<sub>0</sub>, ''v''<sub>1</sub>, ..., ''v''<sub>n</sub>, so that the edges are ''v''<sub>i−1</sub>''v''<sub>i</sub> for each ''i'' = 1, 2, ..., ''n''. If a linear graph occurs as a [[Glossary of graph theory#Subgraphs|subgraph]] of another graph, it is a [[Path (graph theory)|path]] in that graph. | |||
* In a [[cycle graph]] of length ''n ≥ 3'', vertices can be named ''v''<sub>1</sub>, ..., ''v''<sub>n</sub> so that the edges are ''v''<sub>i−1</sub>''v<sub>i</sub>'' for each ''i'' = 2,...,''n'' in addition to ''v''<sub>n</sub>''v''<sub>1</sub>. Cycle graphs can be characterized as [[Connectivity (graph theory)|connected]] [[Regular graph|2-regular]] graphs. If a cycle graph occurs as a subgraph of another graph, it is a ''cycle'' or ''circuit'' in that graph. | |||
* A [[planar graph]] is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect (i.e., ''embedded'' in a plane). | |||
* A [[tree (graph theory)|tree]] is a connected graph with no cycles. | |||
* A ''forest'' is a graph with no cycles (i.e. the disjoint union of one or more ''trees''). | |||
More advanced kinds of graphs are: | |||
* The [[Petersen graph]] and its generalizations | |||
* [[Perfect graph]]s | |||
* [[Cograph]]s | |||
* [[Chordal graph]]s | |||
* Other graphs with large [[Graph automorphism|automorphism groups]]: [[Vertex-transitive graph|vertex-transitive]], [[Arc-transitive graph|arc-transitive]], and [[distance-transitive graph]]s. | |||
* [[Strongly regular graph]]s and their generalization [[distance-regular graph]]s. | |||
==Operations on graphs== | |||
{{main|Operations on graphs}} | |||
There are several operations that produce new graphs from old ones, which might be classified into the following categories: | |||
* Elementary operations, sometimes called "editing operations" on graphs, which create a new graph from the original one by a simple, local change, such as addition or deletion of a vertex or an edge, merging and splitting of vertices, etc. | |||
* [[graph rewriting|Graph rewrite operations]] replacing the occurrence of some pattern graph within the host graph by an instance of the corresponding replacement graph. | |||
* Unary operations, which create a significantly new graph from the old one. Examples: | |||
** [[Line graph]] | |||
** [[Dual graph]] | |||
** [[Complement graph]] | |||
* Binary operations, which create new graph from two initial graphs. Examples: | |||
** [[Disjoint union of graphs]] | |||
** [[Cartesian product of graphs]] | |||
** [[Tensor product of graphs]] | |||
** [[Strong product of graphs]] | |||
** [[Lexicographic product of graphs]] | |||
==Generalizations== | |||
In a [[hypergraph]], an edge can join more than two vertices. | |||
An undirected graph can be seen as a [[simplicial complex]] consisting of 1-[[simplex|simplices]] (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices. | |||
Every graph gives rise to a [[matroid]]. | |||
In [[model theory]], a graph is just a [[structure (model theory)|structure]]. But in that case, there is no limitation on the number of edges: it can be any [[cardinal number]], see [[continuous graph]]. | |||
In [[computational biology]], [[power graph analysis]] introduces power graphs as an alternative representation of undirected graphs. | |||
In [[geographic information systems]], [[geometric networks]] are closely modeled after graphs, and borrow many concepts from [[graph theory]] to perform spatial analysis on road networks or utility grids. | |||
==See also== | |||
* [[Dual graph]] | |||
* [[Glossary of graph theory]] | |||
* [[Hypergraph]] | |||
* [[Graph (data structure)]] | |||
* [[Graph database]] | |||
* [[Graph drawing]] | |||
* [[List of publications in mathematics#Graph theory|Graph theory publications]] | |||
* [[List of graph theory topics]] | |||
* [[Network theory]] | |||
* [[Bayesian network]] | |||
* [[Webgraph]] | |||
* [[Conceptual graph]] | |||
* [[Horizontal constraint graph]] | |||
* [[Causal dynamical triangulation]] | |||
* [[Sage Math]] (software) | |||
* [[NetworkX]] (software) | |||
* [[Mathematica]] (software) | |||
*[[Graph Theory]] | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
* {{cite book|last=Balakrishnan|first=V. K.|title=Graph Theory|publisher=McGraw-Hill|date=1997-02-01|edition=1st|isbn=0-07-005489-4}} | |||
* {{cite book|last=Berge|first=Claude|title=Théorie des graphes et ses applications|publisher=Collection Universitaire de Mathématiques, II|location=Dunod, Paris|year=1958|pages=viii+277|language=French|authorlink=Claude Berge}} Translation: {{cite book|publisher=Wiley|location=Dover, New York|year=2001|origyear=1962|title=-}} | |||
* {{cite book|last=Biggs|first=Norman|title=Algebraic Graph Theory|publisher=Cambridge University Press|year=1993|edition=2nd|isbn=0-521-45897-8}} | |||
* {{cite book|last=Bollobás|first=Béla|title=Modern Graph Theory|publisher=Springer|date=2002-08-12|edition=1st|isbn=0-387-98488-7}} | |||
* {{cite book|last=Bang-Jensen|first=J.|coauthors=Gutin, G.|title=Digraphs: Theory, Algorithms and Applications|publisher=Springer|year=2000|url=http://www.cs.rhul.ac.uk/books/dbook/}} | |||
* {{Cite book | last1=Diestel | first1=Reinhard | title=Graph Theory | url=http://diestel-graph-theory.com/GrTh.html | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-3-540-26183-4 | year=2005 | postscript=<!--None--> }}. | |||
* {{cite book|title=Handbook of Combinatorics|editor=Graham, R.L., Grötschel, M., and Lovász, L|publisher=MIT Press|year=1995|isbn=0-262-07169-X}} | |||
* {{cite book|last=Gross|first=Jonathan L.|coauthors=Yellen, Jay|title=Graph Theory and Its Applications|publisher=CRC Press|date=1998-12-30|isbn=0-8493-3982-0}} | |||
* {{cite book|title=Handbook of Graph Theory|editor=Gross, Jonathan L., & Yellen, Jay|publisher=CRC|date=2003-12-29|isbn=1-58488-090-2}} | |||
* {{cite book|last=Harary|first=Frank|title=Graph Theory|publisher=Addison Wesley Publishing Company|date=January 1995|isbn=0-201-41033-8}} | |||
* {{cite book|last=Iyanaga|first=Shôkichi|coauthors=Kawada, Yukiyosi|title=Encyclopedic Dictionary of Mathematics|publisher=MIT Press|year=1977|isbn=0-262-09016-3}} | |||
* {{cite book|last=Zwillinger|first=Daniel|title=CRC Standard Mathematical Tables and Formulae|publisher=Chapman & Hall/CRC|date=2002-11-27|edition=31st|isbn=1-58488-291-3}} | |||
==Further reading== | |||
*{{cite book|last=Trudeau|first=Richard J.|title=Introduction to Graph Theory|year=1993|publisher=[[Dover Publications]]|location=New York|isbn=978-0-486-67870-2|url=http://store.doverpublications.com/0486678709.html|edition=Corrected, enlarged republication.|accessdate=8 August 2012}} | |||
==External links== | |||
{{Library resources box | |||
|by=no | |||
|onlinebooks=no | |||
|others=no | |||
|about=yes | |||
|label=Graph(mathematics)}} | |||
* {{MathWorld | urlname=Graph | title = Graph}} | |||
{{DEFAULTSORT:Graph (Mathematics)}} | |||
[[Category:Graph theory]] | |||
[[Category:Graph theory objects]] |
Revision as of 17:23, 25 October 2013
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In mathematics, and more specifically in graph theory, a graph is a representation of a set of objects where some pairs of objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges.[1] Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.
The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this is an undirected graph, because if person A shook hands with person B, then person B also shook hands with person A. In contrast, if there is an edge from person A to person B when person A knows of person B, then this graph is directed, because knowledge of someone is not necessarily a symmetric relation (that is, one person knowing another person does not necessarily imply the reverse; for example, many fans may know of a celebrity, but the celebrity is unlikely to know of all their fans). This latter type of graph is called a directed graph and the edges are called directed edges or arcs.
Vertices are also called nodes or points, and edges are also called arcs or lines. Graphs are the basic subject studied by graph theory. The word "graph" was first used in this sense by J.J. Sylvester in 1878.[2][3]
Definitions
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures.
Graph
In the most common sense of the term,[4] a graph is an ordered pair G = (V, E) comprising a set V of vertices or nodes together with a set E of edges or lines, which are 2-element subsets of V (i.e., an edge is related with two vertices, and the relation is represented as an unordered pair of the vertices with respect to the particular edge). To avoid ambiguity, this type of graph may be described precisely as undirected and simple.
Other senses of graph stem from different conceptions of the edge set. In one more generalized notion,[5] E is a set together with a relation of incidence that associates with each edge two vertices. In another generalized notion, E is a multiset of unordered pairs of (not necessarily distinct) vertices. Many authors call this type of object a multigraph or pseudograph.
All of these variants and others are described more fully below.
The vertices belonging to an edge are called the ends, endpoints, or end vertices of the edge. A vertex may exist in a graph and not belong to an edge.
V and E are usually taken to be finite, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the infinite case. The order of a graph is (the number of vertices). A graph's size is , the number of edges. The degree of a vertex is the number of edges that connect to it, where an edge that connects to the vertex at both ends (a loop) is counted twice.
For an edge {u, v}, graph theorists usually use the somewhat shorter notation uv.
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Adjacency relation
The edges E of an undirected graph G induce a symmetric binary relation ~ on V that is called the adjacency relation of G. Specifically, for each edge {u, v} the vertices u and v are said to be adjacent to one another, which is denoted u ~ v.
Types of graphs
Distinction in terms of the main definition
As stated above, in different contexts it may be useful to define the term graph with different degrees of generality. Whenever it is necessary to draw a strict distinction, the following terms are used. Most commonly, in modern texts in graph theory, unless stated otherwise, graph means "undirected simple finite graph" (see the definitions below).
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Undirected graph
An undirected graph is one in which edges have no orientation. The edge (a, b) is identical to the edge (b, a), i.e., they are not ordered pairs, but sets {u, v} (or 2-multisets) of vertices. The maximum number of edges in an undirected graph without a self-loop is n(n - 1)/2.
Directed graph
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A directed graph or digraph is an ordered pair D = (V, A) with
- V a set whose elements are called vertices or nodes, and
- A a set of ordered pairs of vertices, called arcs, directed edges, or arrows.
An arc a = (x, y) is considered to be directed from x to y; y is called the head and x is called the tail of the arc; y is said to be a direct successor of x, and x is said to be a direct predecessor of y. If a path leads from x to y, then y is said to be a successor of x and reachable from x, and x is said to be a predecessor of y. The arc (y, x) is called the arc (x, y) inverted.
A directed graph D is called symmetric if, for every arc in D, the corresponding inverted arc also belongs to D. A symmetric loopless directed graph D = (V, A) is equivalent to a simple undirected graph G = (V, E), where the pairs of inverse arcs in A correspond 1-to-1 with the edges in E; thus the edges in G number |E| = |A|/2, or half the number of arcs in D.
A variation on this definition is the oriented graph, in which at most one of (x, y) and (y, x) may be arcs.
Mixed graph
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A mixed graph G is a graph in which some edges may be directed and some may be undirected. It is written as an ordered triple G = (V, E, A) with V, E, and A defined as above. Directed and undirected graphs are special cases.
Multigraph
A loop is an edge (directed or undirected) which starts and ends on the same vertex; these may be permitted or not permitted according to the application. In this context, an edge with two different ends is called a link.
The term "multigraph" is generally understood to mean that multiple edges (and sometimes loops) are allowed. Where graphs are defined so as to allow loops and multiple edges, a multigraph is often defined to mean a graph without loops,[6] however, where graphs are defined so as to disallow loops and multiple edges, the term is often defined to mean a "graph" which can have both multiple edges and loops,[7] although many use the term "pseudograph" for this meaning.[8]
Quiver
A quiver or "multidigraph" is a directed graph which may have more than one arrow from a given source to a given target. A quiver may also have directed loops in it.
Simple graph
As opposed to a multigraph, a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices. In a simple graph the edges of the graph form a set (rather than a multiset) and each edge is a distinct pair of vertices. In a simple graph with n vertices every vertex has a degree that is less than n (the converse, however, is not true — there exist non-simple graphs with n vertices in which every vertex has a degree smaller than n).
Weighted graph
A graph is a weighted graph if a number (weight) is assigned to each edge.[9] Such weights might represent, for example, costs, lengths or capacities, etc. depending on the problem at hand. Some authors call such a graph a network.[10] Weighted correlation networks can be defined by soft-thresholding the pairwise correlations among variables (e.g. gene measurements).
Half-edges, loose edges
In certain situations it can be helpful to allow edges with only one end, called half-edges, or no ends (loose edges); see for example signed graphs and biased graphs.
Important graph classes
Regular graph
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. A regular graph is a graph where each vertex has the same number of neighbours, i.e., every vertex has the same degree or valency. A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k.
Complete graph
![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Complete_graph_K5.svg/125px-Complete_graph_K5.svg.png)
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Complete graphs have the feature that each pair of vertices has an edge connecting them.
Finite and infinite graphs
A finite graph is a graph G = (V, E) such that V and E are finite sets. An infinite graph is one with an infinite set of vertices or edges or both.
Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated.
Graph classes in terms of connectivity
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In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v. Otherwise, they are called disconnected. A graph is called connected if every pair of distinct vertices in the graph is connected; otherwise, it is called disconnected.
A graph is called k-vertex-connected or k-edge-connected if no set of k-1 vertices (respectively, edges) exists that, when removed, disconnects the graph. A k-vertex-connected graph is often called simply k-connected.
A directed graph is called weakly connected if replacing all of its directed edges with undirected edges produces a connected (undirected) graph. It is strongly connected or strong if it contains a directed path from u to v and a directed path from v to u for every pair of vertices u, v.
Category of all graphs
The category of all graphs is the slice category where is the functor taking a set to .
Properties of graphs
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Two edges of a graph are called adjacent if they share a common vertex. Two arrows of a directed graph are called consecutive if the head of the first one is at the nock (notch end) of the second one. Similarly, two vertices are called adjacent if they share a common edge (consecutive if they are at the notch and at the head of an arrow), in which case the common edge is said to join the two vertices. An edge and a vertex on that edge are called incident.
The graph with only one vertex and no edges is called the trivial graph. A graph with only vertices and no edges is known as an edgeless graph. The graph with no vertices and no edges is sometimes called the null graph or empty graph, but the terminology is not consistent and not all mathematicians allow this object.
In a weighted graph or digraph, each edge is associated with some value, variously called its cost, weight, length or other term depending on the application; such graphs arise in many contexts, for example in optimal routing problems such as the traveling salesman problem.
Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called vertex-labeled. However, for many questions it is better to treat vertices as indistinguishable; then the graph may be called unlabeled. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges). The same remarks apply to edges, so graphs with labeled edges are called edge-labeled graphs. Graphs with labels attached to edges or vertices are more generally designated as labeled. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called unlabeled. (Note that in the literature the term labeled may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.)
Examples
![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5b/6n-graf.svg/300px-6n-graf.svg.png)
- The diagram at right is a graphic representation of the following graph:
- V = {1, 2, 3, 4, 5, 6}
- E = {{1, 2}, {1, 5}, {2, 3}, {2, 5}, {3, 4}, {4, 5}, {4, 6}}.
- In category theory a small category can be represented by a directed multigraph in which the objects of the category represented as vertices and the morphisms as directed edges. Then, the functors between categories induce some, but not necessarily all, of the digraph morphisms of the graph.
- In computer science, directed graphs are used to represent knowledge (e.g., Conceptual graph), finite state machines, and many other discrete structures.
- A binary relation R on a set X defines a directed graph. An element x of X is a direct predecessor of an element y of X iff xRy.
Important graphs
Basic examples are:
- In a complete graph, each pair of vertices is joined by an edge; that is, the graph contains all possible edges.
- In a bipartite graph, the vertex set can be partitioned into two sets, W and X, so that no two vertices in W are adjacent and no two vertices in X are adjacent. Alternatively, it is a graph with a chromatic number of 2.
- In a complete bipartite graph, the vertex set is the union of two disjoint sets, W and X, so that every vertex in W is adjacent to every vertex in X but there are no edges within W or X.
- In a linear graph or path graph of length n, the vertices can be listed in order, v0, v1, ..., vn, so that the edges are vi−1vi for each i = 1, 2, ..., n. If a linear graph occurs as a subgraph of another graph, it is a path in that graph.
- In a cycle graph of length n ≥ 3, vertices can be named v1, ..., vn so that the edges are vi−1vi for each i = 2,...,n in addition to vnv1. Cycle graphs can be characterized as connected 2-regular graphs. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph.
- A planar graph is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect (i.e., embedded in a plane).
- A tree is a connected graph with no cycles.
- A forest is a graph with no cycles (i.e. the disjoint union of one or more trees).
More advanced kinds of graphs are:
- The Petersen graph and its generalizations
- Perfect graphs
- Cographs
- Chordal graphs
- Other graphs with large automorphism groups: vertex-transitive, arc-transitive, and distance-transitive graphs.
- Strongly regular graphs and their generalization distance-regular graphs.
Operations on graphs
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There are several operations that produce new graphs from old ones, which might be classified into the following categories:
- Elementary operations, sometimes called "editing operations" on graphs, which create a new graph from the original one by a simple, local change, such as addition or deletion of a vertex or an edge, merging and splitting of vertices, etc.
- Graph rewrite operations replacing the occurrence of some pattern graph within the host graph by an instance of the corresponding replacement graph.
- Unary operations, which create a significantly new graph from the old one. Examples:
- Binary operations, which create new graph from two initial graphs. Examples:
Generalizations
In a hypergraph, an edge can join more than two vertices.
An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.
Every graph gives rise to a matroid.
In model theory, a graph is just a structure. But in that case, there is no limitation on the number of edges: it can be any cardinal number, see continuous graph.
In computational biology, power graph analysis introduces power graphs as an alternative representation of undirected graphs.
In geographic information systems, geometric networks are closely modeled after graphs, and borrow many concepts from graph theory to perform spatial analysis on road networks or utility grids.
See also
- Dual graph
- Glossary of graph theory
- Hypergraph
- Graph (data structure)
- Graph database
- Graph drawing
- Graph theory publications
- List of graph theory topics
- Network theory
- Bayesian network
- Webgraph
- Conceptual graph
- Horizontal constraint graph
- Causal dynamical triangulation
- Sage Math (software)
- NetworkX (software)
- Mathematica (software)
- Graph Theory
Notes
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References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Translation: 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534. - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
Further reading
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
External links
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- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ See:
- J. J. Sylvester (February 7, 1878) "Chemistry and algebra," Nature, 17 : 284. From page 284: "Every invariant and covariant thus becomes expressible by a graph precisely identical with a Kekuléan diagram or chemicograph."
- J. J. Sylvester (1878) "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, — with three appendices," American Journal of Mathematics, Pure and Applied, 1 (1) : 64-90. The term "graph" first appears in this paper on page 65.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ See, for instance, Iyanaga and Kawada, 69 J, p. 234 or Biggs, p. 4.
- ↑ See, for instance, Graham et al., p. 5.
- ↑ For example, see Balakrishnan, p. 1, Gross (2003), p. 4, and Zwillinger, p. 220.
- ↑ For example, see. Bollobás, p. 7 and Diestel, p. 25.
- ↑ Gross (1998), p. 3, Gross (2003), p. 205, Harary, p.10, and Zwillinger, p. 220.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 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