Editing Graph (discrete mathematics) (section)

=== {{anchor|Undirected graph}} Graph ===
[[File:Undirected.svg|thumb|upright|A graph with three vertices and three edges]]

A '''graph''' (sometimes called an ''undirected graph'' to distinguish it from a [[#Directed graph|directed graph]], or a ''simple graph'' to distinguish it from a [[multigraph]]){{sfn|Bender|Williamson|2010|p=148}}<ref>See, for instance, Iyanaga and Kawada, ''69 J'', p. 234 or Biggs, p. 4.</ref> is a [[ordered pair|pair]] {{math|1=''G'' = (''V'', ''E'')}}, where {{mvar|V}} is a set whose elements are called ''vertices'' (singular: vertex), and {{mvar|E}} is a set of unordered pairs <math>\{v_1, v_2\}</math> of vertices, whose elements are called ''edges'' (sometimes ''links'' or ''lines'').

The vertices {{mvar|u}} and {{mvar|v}} of an edge {{math|{''u'', ''v''} }} are called the edge's ''endpoints''. The edge is said to ''join'' {{mvar|u}} and {{mvar|v}} and to be ''incident'' on them. A vertex may belong to no edge, in which case it is not joined to any other vertex and is called ''isolated''. When an edge <math>\{u,v\}</math> exists, the vertices {{mvar|u}} and {{mvar|v}} are called ''adjacent''.

A [[multigraph]] is a generalization that allows multiple edges to have the same pair of endpoints. In some texts, multigraphs are simply called graphs.{{sfn|Bender|Williamson|2010|p=149}}<ref>Graham et al., p. 5.</ref>

Sometimes, graphs are allowed to contain ''[[Loop (graph theory)|loop]]s'', which are edges that join a vertex to itself. To allow loops, the pairs of vertices in {{mvar|E}} must be allowed to have the same node twice.  Such generalized graphs are called ''graphs with loops'' or simply ''graphs'' when it is clear from the context that loops are allowed.

Generally, the vertex set {{mvar|V}} is taken to be finite (which implies that the edge set {{mvar|E}} is also finite). Sometimes [[infinite graph]]s are considered, but they are usually viewed as a special kind of [[binary relation]], because most results on finite graphs either do not extend to the infinite case or need a rather different proof.

An [[empty graph]] is a graph that has an [[empty set]] of vertices (and thus an empty set of edges). The ''order'' of a graph is its number {{math|{{abs|''V''}}}} of vertices, usually denoted by {{mvar|n}}. The ''size'' of a graph is its number {{math|{{abs|''E''}}}} of edges, typically denoted by {{mvar|m}}. However, in some contexts, such as for expressing the [[computational complexity]] of algorithms, the term ''size'' is used for the quantity {{math|{{abs|''V''}} + {{abs|''E''}}}} (otherwise, a non-empty graph could have size 0). The ''degree'' or ''valency'' of a vertex is the number of edges that are incident to it; for graphs with loops, a loop is counted twice.

In a graph of order {{math|''n''}}, the maximum degree of each vertex is {{math|''n'' − 1}} (or {{math|''n'' + 1}} if loops are allowed, because a loop contributes 2 to the degree), and the maximum number of edges is {{math|''n''(''n'' − 1)/2}} (or {{math|''n''(''n'' + 1)/2}} if loops are allowed).

The edges of a graph define a [[symmetric relation]] on the vertices, called the ''adjacency relation''. Specifically, two vertices {{mvar|x}} and {{mvar|y}} are ''adjacent'' if {{math|{''x'', ''y''} }} is an edge. A graph is fully determined by its [[adjacency matrix]] {{mvar|A}}, which is an {{math|''n'' × ''n''}} square matrix, with {{mvar|A{{sub|ij}}}} specifying the number of connections from vertex {{mvar|i}} to vertex {{mvar|j}}. For a simple graph, {{math|''A{{sub|ij}}''}} is either 0, indicating disconnection, or 1, indicating connection; moreover {{math|1=''A{{sub|ii}}'' = 0}} because an edge in a simple graph cannot start and end at the same vertex. Graphs with self-loops will be characterized by some or all {{mvar|A{{sub|ii}}}} being equal to a positive integer, and multigraphs (with multiple edges between vertices) will be characterized by some or all {{mvar|A{{sub|ij}}}} being equal to a positive integer. Undirected graphs will have a [[symmetric matrix|symmetric]] adjacency matrix (meaning {{math|1=''A{{sub|ij}}'' = ''A{{sub|ji}}''}}).