Graph (mathematics)

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, a graph is an abstract representation of a set of objects where some pairs of the 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. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges.

For example, a graph may be constructed by choosing the vertices to be the first 1000 positive integers, and defining that there is an edge between two vertices if and only if those two integers have at least one decimal digit in common.

In other cases the relationship between vertices is not symmetric: for example, a graph may be constructed by choosing the vertices to be the first 1000 positive integers, and defining that there is an edge from i to j if i is a divisor of j.

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, a graph is an abstract representation of a set of objects where some pairs of the 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. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges.

For example, a graph may be constructed by choosing the vertices to be the first 1000 positive integers, and defining that there is an edge between two vertices if and only if those two integers have at least one decimal digit in common.

In other cases the relationship between vertices is not symmetric: for example, a graph may be constructed by choosing the vertices to be the first 1000 positive integers, and defining that there is an edge from i to j if i is a divisor of j. This type of graph is called a directed graph and the edges are called directed edges or arcs; in contrast, a graph where the edges are not directed is called undirected.

Vertices are also called nodes or points, and edges are also called lines. Graphs are the basic subject studied by graph theory

Graph theory

In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

.

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,
a graph is an ordered pair

Ordered pair

In mathematics, an ordered pair is a collection of objects having two coordinates , such that one can always uniquely determine the object, which is the first coordinate of the pair as well as the second coordinate...

  comprising a set  of vertices or nodes together with a set of edges or lines, which are 2-element subsets of . 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, is a set together with a relation of incidence that associates with each edge two vertices. In another generalized notion, is a multiset

Multiset

In mathematics, a multiset is a generalization of a set. While each member of a set has only one membership, a member of a multiset can have more than one membership...

 of unordered pairs of (not necessarily distinct) vertices. Many authors call this type of object a multigraph

Multigraph

In mathematics, a multigraph or pseudograph is a graph which is permitted to have multiple edges, , that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge...

 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.

and 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

Loop (graph theory)

In graph theory, a loop is an edge that connects a vertex to itself. A simple graph contains no loops.Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops :*Where graphs are defined so as to...

) is counted twice.

The edges of an undirected graph induce a symmetric binary relation ~ on that is called the adjacency

Adjacency matrix

In mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices...

 relation of . Specifically, for each edge {u,v} the vertices u and v are said to be adjacent to one another, which is denoted u ~ v.

For an edge {u, v}, graph theorists usually use the somewhat shorter notation uv.

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).

Undirected graph

A graph in which edges have no orientation, i.e., they are not ordered pairs, but sets {u, v} (or 2-multisets) of vertices.

Directed graph

A directed graph or digraph is an ordered pair with

• a set whose elements are called vertices or nodes, and
• a set of ordered pairs of vertices, called arcs, directed edges, or arrows.

An arc is considered to be directed from to ; is called the head and is called the tail of the arc; is said to be a direct successor of , and is said to be a direct predecessor of . If a path

Path (graph theory)

In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...

 leads from to , then is said to be a successor of and reachable from , and is said to be a predecessor of . The arc $(y,\; x)$ is called the arc

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, a graph is an abstract representation of a set of objects where some pairs of the 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. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges.

For example, a graph may be constructed by choosing the vertices to be the first 1000 positive integers, and defining that there is an edge between two vertices if and only if those two integers have at least one decimal digit in common.

In other cases the relationship between vertices is not symmetric: for example, a graph may be constructed by choosing the vertices to be the first 1000 positive integers, and defining that there is an edge from i to j if i is a divisor of j. This type of graph is called a directed graph and the edges are called directed edges or arcs; in contrast, a graph where the edges are not directed is called undirected.

Vertices are also called nodes or points, and edges are also called lines. Graphs are the basic subject studied by graph theory

Graph theory

In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

.

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,
a graph is an ordered pair

Ordered pair

In mathematics, an ordered pair is a collection of objects having two coordinates , such that one can always uniquely determine the object, which is the first coordinate of the pair as well as the second coordinate...

 $G\; :=\; (V,\; E)$ comprising a set $V$ of vertices or nodes together with a set $E$ of edges or lines, which are 2-element subsets of $V$. 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, $E$ is a set together with a relation of incidence that associates with each edge two vertices. In another generalized notion, $E$ is a multiset

Multiset

In mathematics, a multiset is a generalization of a set. While each member of a set has only one membership, a member of a multiset can have more than one membership...

 of unordered pairs of (not necessarily distinct) vertices. Many authors call this type of object a multigraph

Multigraph

In mathematics, a multigraph or pseudograph is a graph which is permitted to have multiple edges, , that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge...

 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 $|V|$ (the number of vertices). A graph's size is $|E|$, 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

Loop (graph theory)

In graph theory, a loop is an edge that connects a vertex to itself. A simple graph contains no loops.Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops :*Where graphs are defined so as to...

) is counted twice.

The edges $E$ of an undirected graph $G$ induce a symmetric binary relation ~ on $V$ that is called the adjacency

Adjacency matrix

In mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices...

 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.

For an edge {u, v}, graph theorists usually use the somewhat shorter notation uv.

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).

Undirected graph

A graph in which edges have no orientation, i.e., they are not ordered pairs, but sets {u, v} (or 2-multisets) of vertices.

Directed graph

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

Path (graph theory)

In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...

 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
In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, a graph is an abstract representation of a set of objects where some pairs of the 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. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges.

For example, a graph may be constructed by choosing the vertices to be the first 1000 positive integers, and defining that there is an edge between two vertices if and only if those two integers have at least one decimal digit in common.

In other cases the relationship between vertices is not symmetric: for example, a graph may be constructed by choosing the vertices to be the first 1000 positive integers, and defining that there is an edge from i to j if i is a divisor of j. This type of graph is called a directed graph and the edges are called directed edges or arcs; in contrast, a graph where the edges are not directed is called undirected.

Vertices are also called nodes or points, and edges are also called lines. Graphs are the basic subject studied by graph theory

Graph theory

In mathematics and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

.

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,
a graph is an ordered pair

Ordered pair

In mathematics, an ordered pair is a collection of objects having two coordinates , such that one can always uniquely determine the object, which is the first coordinate of the pair as well as the second coordinate...

 $G\; :=\; (V,\; E)$ comprising a set $V$ of vertices or nodes together with a set $E$ of edges or lines, which are 2-element subsets of $V$. 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, $E$ is a set together with a relation of incidence that associates with each edge two vertices. In another generalized notion, $E$ is a multiset

Multiset

In mathematics, a multiset is a generalization of a set. While each member of a set has only one membership, a member of a multiset can have more than one membership...

 of unordered pairs of (not necessarily distinct) vertices. Many authors call this type of object a multigraph

Multigraph

In mathematics, a multigraph or pseudograph is a graph which is permitted to have multiple edges, , that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge...

 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 $|V|$ (the number of vertices). A graph's size is $|E|$, 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

Loop (graph theory)

In graph theory, a loop is an edge that connects a vertex to itself. A simple graph contains no loops.Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops :*Where graphs are defined so as to...

) is counted twice.

The edges $E$ of an undirected graph $G$ induce a symmetric binary relation ~ on $V$ that is called the adjacency

Adjacency matrix

In mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices...

 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.

For an edge {u, v}, graph theorists usually use the somewhat shorter notation uv.

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).

Undirected graph

A graph in which edges have no orientation, i.e., they are not ordered pairs, but sets {u, v} (or 2-multisets) of vertices.

Directed graph

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

Path (graph theory)

In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...

 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 not more than one of $(x,\; y)$ and $(y,\; x)$ may be arcs.

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

Loop (graph theory)

In graph theory, a loop is an edge that connects a vertex to itself. A simple graph contains no loops.Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops :*Where graphs are defined so as to...

 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

Multigraph

In mathematics, a multigraph or pseudograph is a graph which is permitted to have multiple edges, , that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge...

" 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, 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, although many use the term "pseudograph" for this meaning.

Simple graph

As opposed to a multigraph, a simple graph is an undirected graph that has no loops

Loop (graph theory)

In graph theory, a loop is an edge that connects a vertex to itself. A simple graph contains no loops.Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops :*Where graphs are defined so as to...

 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

Multiset

In mathematics, a multiset is a generalization of a set. While each member of a set has only one membership, a member of a multiset can have more than one membership...

) and each edge is a pair of distinct 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. Such weights might represent, for example, costs, lengths or capacities, etc. depending on the problem.

The weight of the graph is sum of the weights given to all edges.

Half-edges, loose edges

In exceptional situations it is even necessary to have edges with only one end, called half-edges, or no ends (loose edges); see for example signed graph

Signed graph

In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign.Formally, a signed graph Σ is a pair that consists of a graph G = and a sign mapping or signature σ from E to the sign group {+,−}...

s and biased graph

Biased graph

In mathematics, a biased graph is a graph with a list of distinguished circles , such that if two circles in the list are contained in a theta graph, then so is the third circle of the theta graph...

s.

Regular graph

A regular graph is a graph where each vertex has the same number of neighbors, 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

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(G) and E(G) are finite set

Finite set

In mathematics, finite set is a set that has a finite number of elements. For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set. A set that is not finite is called infinite...

s. An infinite graph is the one with sets of vertices or edges or both infinite

Infinite set

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:* the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and...

.

Most commonly in graph theory it is implied that the discussed graphs are finite, i.e., finite graphs are called simply "graphs", while the infinite graphs are called so in full.

Graph classes in terms of connectivity

In an undirected graph G, two vertices

Vertex (graph theory)

In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...

 u and v are called connected if G contains a path

Path (graph theory)

In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...

 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 and disconnected otherwise.

A graph is called k-vertex-connected or k-edge-connected if removal of k or more vertices (respectively, edges) makes the graph disconnected. A k-vertex-connected graph is often called simply k-connected.

A directed graph

Directed graph

A directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,* a set A of ordered pairs of vertices, called arcs, directed edges, or arrows .It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of...

 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.

Properties of graphs

Two edges of a graph are called adjacent (sometimes coincident) 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

Nock

Nock may refer to:* Nock - the notch in the end of an arrow* Nock - to mount an arrow to a bow * Nock - members of the Nock family of gunsmiths in England**Henry Nock - gunsmith who also founded Wilkinson Sword in 1772...

 (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 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 that graphs which have 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

The picture is a graphic representation of the following graph

• $V\; :=\; \backslash \{1,\; 2,\; 3,\; 4,\; 5,\; 6\backslash \}$
• $E\; :=\; \backslash \{\backslash \{1,\; 2\backslash \},\; \backslash \{1,\; 5\backslash \},\; \backslash \{2,\; 3\backslash \},\; \backslash \{2,\; 5\backslash \},\; \backslash \{3,\; 4\backslash \},\; \backslash \{4,\; 5\backslash \},\; \backslash \{4,\; 6\backslash \}\backslash \}.$

The fact that vertex 1 is adjacent to vertex 2 is sometimes denoted by 1 ~ 2.

• In category theory
  Category theory
  In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
  
   a small category
  Category (mathematics)
  In mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may...
  
   can be considered a directed multigraph
  Multigraph
  In mathematics, a multigraph or pseudograph is a graph which is permitted to have multiple edges, , that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge...
  
   with the objects as vertices and the morphism
  Morphism
  In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory...
  
  s as directed edges. The functor
  Functor
  In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms in the category of small categories....
  
  s between categories induce then some, but not necessarily all, of the digraph morphisms.
• In computer science
  Computer science
  Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe and transform...
  
   directed graphs are used to represent finite state machine
  Finite state machine
  A finite state machine or finite state automaton , or simply a state machine, is a model of behavior composed of a finite number of states, transitions between those states, and actions. It is similar to a "flow graph" where we can inspect the way in which the logic runs when certain conditions...
  
  s and many other discrete structures.
• A binary relation
  Binary relation
  In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A² = . More generally, a binary relation between two sets A and B is a subset of...
  
   $R$ on a set $X$ is a directed graph. Two edges $x$, $y$ of $X$ are connected by an arrow if $xRy$.

Important graphs

Basic examples are:

• In a complete graph
  Complete graph
  In the mathematical field of graph theory, a complete graph is a simple graph in which every pair of distinct vertices is connected by an edge. The complete graph on n vertices has n vertices and n/2 edges, and is denoted by . It is a regular graph of degree . All complete graphs are their own...
  
   each pair of vertices is joined by an edge, that is, the graph contains all possible edges.
• In a bipartite graph
  Bipartite graph
  In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V are independent sets...
  
  , the vertices can be divided into two sets, W and X, so that every edge has one vertex in each of the two sets.
• In a complete bipartite graph
  Complete bipartite graph
  In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.- Definition :...
  
  , the vertex set is the union of two disjoint subsets, 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 path
  Path (graph theory)
  In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...
  
   of length n, the vertices can be listed in order, v₀, v₁, ..., v_n, so that the edges are v_i−1v_i for each i = 1, 2, ..., n.
• A cycle
  Cycle graph
  In graph theory, a cycle graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called C_n...
  
   or circuit of length n is a closed path without self-intersections; equivalently, it is a connected
  Connectivity (graph theory)
  In mathematics and computer science, connectivity is one of the basic concepts of graph theory. It is closely related to the theory of network flow problems...
  
   graph with degree 2 at every vertex. Its vertices can be named v₁, ..., v_n so that the edges are v_i−1vi for each i = 2,...,n and v_nv₁
• A planar graph
  Planar graph
  In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints....
  
   can be drawn in a plane with no crossing edges (i.e., embedded in a plane).
• A tree
  Tree (graph theory)
  In mathematics, more specifically graph theory, a tree is a graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...
  
   is a connected graph with no cycles.
• A forest is a graph with no cycles (i.e. one or more trees).

More advanced kinds of graphs are:

• The Petersen graph
  Petersen graph
  In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named for Julius Petersen, who in 1898 constructed it...
  
   and its generalizations
• Perfect graph
  Perfect graph
  In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph....
  
  s
• Cograph
  Cograph
  In graph theory, a cograph, or complement-reducible graph , or P₄-free graph, is a graph that can be generated from the single-vertex graph K₁ by complementation and disjoint union...
  
  s
• Other graphs with large automorphism groups
  Graph automorphism
  In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity....
  
  : vertex-transitive
  Vertex-transitive graph
  In the mathematical field of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v₁ and v₂ of G, there is some automorphismsuch that...
  
  , arc-transitive, and distance-transitive graph
  Distance-transitive graph
  In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance, there is an automorphism of the graph that carries v to x and w to y.A distance transitive...
  
  s.
• Strongly regular graph
  Strongly regular graph
  Let G = be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers λ and μ such that:* Every two adjacent vertices have λ common neighbours....
  
  s and their generalization distance-regular graph
  Distance-regular graph
  In mathematics, a distance-regular graph is a regular graph such that for any two vertices v and w at distance i the number of vertices adjacent to w and at distance j from v is the same. Every distance-transitive graph is distance regular...
  
  s.

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 rewrite operations
  Graph rewriting
  Graph transformation, or Graph rewriting, concerns the technique to create a new graph out of an original graph using some automatic machine. It has numerous applications, ranging from software verification to layout algorithms....
  
   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
    Line graph
    In a graph theory, the line graph L of an undirected graph G is another graph L that represents the adjacencies between edges of G...
    
    
  • Dual graph
    Dual graph
    In mathematics, a dual graph of a given planar graph G is a graph which has a vertex for each plane region of G, and an edge for each edge in G joining two neighboring regions, for a certain embedding of G. The term "dual" is used because this property is symmetric, meaning that if H is a dual of...
    
    
  • Complement graph
    Complement graph
    In graph theory, the complement or inverse of a graph G is a graph H on the same vertices such that two vertices of H are adjacent if and only if they are not adjacent in G. That is to find the complement of a graph, you fill in all the missing edges to get a complete graph, and remove all the...
    
    
• Binary operations, which create new graph from two initial graphs. Examples:
  • Disjoint union of graphs
  • Cartesian product of graphs
    Cartesian product of graphs
    In graph theory, the Cartesian product G H of graphs G and H is a graph such that* the vertex set of G H is the Cartesian product V × V; and...
    
    
  • Tensor product of graphs
    Tensor product of graphs
    In graph theory, the tensor product G × H of graphs G and H is a graph such that* the vertex set of G × H is the Cartesian product V × V; and...
    
    
  • Strong product of graphs
  • Lexicographic product of graphs
    Lexicographic product of graphs
    In graph theory, the lexicographic product or graph composition G ∙ H of graphs G and H is a graph such that* the vertex set of G ∙ H is the cartesian product V V; and...
    
    

Generalizations

In a hypergraph

Hypergraph

In mathematics, a hypergraph is a generalization of a graph, where edges can connect any number of vertices. Formally, a hypergraph is a pair where is a set of elements, called nodes or vertices, and is a set of non-empty subsets of called hyperedges or links...

, an edge can join more than two vertices.

An undirected graph can be seen as a simplicial complex

Simplicial complex

In mathematics, a simplicial complex is a topological space of a particular kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...

 consisting of 1-simplices

Simplex

In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope with n + 1 vertices, of which the simplex is the convex hull...

 (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

Matroid

In combinatorics, a branch of mathematics, a matroid or independence structure is a structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces....

.

In model theory

Model theory

In mathematics, model theory is the study of mathematical structures such as groups, fields, graphs, or even universes of set theory, using tools from mathematical logic. A structure that gives meaning to the sentences of a formal language is called a model for the language...

, 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

Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

.

In computational biology

Computational biology

Computational biology is an interdisciplinary field that applies the techniques of computer science, applied mathematics and statistics to address biological problems. The main focus lies on developing mathematical modeling and computational simulation techniques...

, power graph analysis

Power graph analysis

In computational biology, power graph analysis is a method for the analysis andrepresentation of complex networks. Power graph analysis is the computation, analysis and visual representation of a power graph from a graphs ....

 introduces power graphs as an alternative representation of undirected graphs.

See also

• Dual graph
  Dual graph
  In mathematics, a dual graph of a given planar graph G is a graph which has a vertex for each plane region of G, and an edge for each edge in G joining two neighboring regions, for a certain embedding of G. The term "dual" is used because this property is symmetric, meaning that if H is a dual of...
  
  
• Glossary of graph theory
  Glossary of graph theory
  Graph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different meanings. Some authors use different words to mean the same thing...
  
  
• Graph (data structure)
  Graph (data structure)
  In computer science, a graph is an abstract data structure that is meant to implement the graph concept from mathematics.A graph data structure consists mainly of a finite set of ordered pairs, called edges or arcs, of certain entities called nodes or vertices...
  
  
• Graph drawing
  Graph drawing
  Graph drawing or Graph layout, as a branch of graph theory, applies topology and geometry to derive two-dimensional representations of graphs...
  
  
• Graph theory publications
• List of graph theory topics
• List of graph theory terms
• Network theory
  Network theory
  Network theory is an area of applied mathematics and network science and part of graph theory. It has application in many disciplines including particle physics, computer science, biology, economics, operations research, and sociology...
  
  
• Horizontal constraint graph
  Horizontal constraint graph
  In constraint satisfaction, a branch of mathematics, a horizontal constraint graph helps visualize horizontal constraints. For instance in a channel routing problem, two nets n_i and n_j are said to have horizontal constraints, if their horizontal spans have at least one column...
  
  

External links

• Graph theory tutorial
• Some graph theory algorithm animations
• Challenging Benchmarks for Maximum Clique, Maximum Independent Set, Minimum Vertex Cover and Vertex Coloring
• Image gallery : Some real-life graphs
• VisualComplexity.com — A visual exploration on mapping complex networks
• Grafos Spanish copyleft software
• Edge Addition Planarity Algorithm — Online version of a paper that describes the Boyer-Myrvold planarity algorithm.
• Edge Addition Planarity Algorithm Source Code — Free C source code for reference implementation of Boyer-Myrvold planarity algorithm, which provides both a combinatorial planar embedder and Kuratowski subgraph isolator.
• Library of Efficient Models and Optimization in Networks — An open source C++ template library aimed at combinatorial optimization tasks, especially those working with graphs and networks.
• TORSCHE Scheduling Toolbox for Matlab — A freely available toolbox of scheduling and graph algorithms.