Graph (discrete mathematics): Difference between revisions

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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’sedge'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>