Graph (discrete mathematics): Difference between revisions

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[[File:6n-graf.svg|thumb|A graph with six vertices and seven edges]]

In [[discrete mathematics]], particularly in [[graph theory]], a '''graph''' is a structure consisting of a [[Set (mathematics)|set]] of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''[[Vertex (graph theory)|vertices]]'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line'').<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.|access-date=8 August 2012|quote=A graph is an object consisting of two sets called its ''vertex set'' and its ''edge set''.|archive-date=5 May 2019|archive-url=https://web.archive.org/web/20190505192352/http://store.doverpublications.com/0486678709.html|url-status=live}}</ref> Typically, a graph is depicted in [[diagrammatic form]] as a set of dots or circles for the vertices, joined by lines or curves for the edges.

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 graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing money is not necessarily reciprocated.

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 due to a direct relation between mathematics and [[chemical structure]] (what he called a chemico-graphical image).<ref>See:

* J. J. Sylvester (February 7, 1878) [https://books.google.com/books?id=KcoKAAAAYAAJ&q=Sylvester&pg=PA284 "Chemistry and algebra"], {{Webarchive|url=https://web.archive.org/web/20230204142956/https://books.google.com/books?id=KcoKAAAAYAAJ&vq=Sylvester&pg=PA284 |date=2023-02-04 }} ''Nature'', ''17'' : 284. {{doi|10.1038/017284a0}}. 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) [https://books.google.com/books?id=1q0EAAAAYAAJ&pg=PA64 "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, – with three appendices"], {{Webarchive|url=https://web.archive.org/web/20230204142957/https://books.google.com/books?id=1q0EAAAAYAAJ&pg=PA64 |date=2023-02-04 }} ''American Journal of Mathematics, Pure and Applied'', ''1'' (1) : 64–90. {{doi|10.2307/2369436}}. {{JSTOR|2369436}}. 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 = [https://books.google.com/books?id=mKkIGIea_BkC&pg=PA35 35]

| isbn = 978-1-58488-090-5

| url = https://books.google.com/books?id=mKkIGIea_BkC

| access-date = 2016-02-16

| archive-date = 2023-02-04

| archive-url = https://web.archive.org/web/20230204142959/https://books.google.com/books?id=mKkIGIea_BkC

| url-status = live

}}</ref>

== Definitions ==

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== Properties of graphs ==

{{see also|Glossary of graph theory|Graph property}}

Two edgesvertices of a graph are called ''adjacent'' if they share a common vertexedge. Two edgesvertices of a directed graph are called ''consecutive'' if the head of the first one is the tail of the second one. Similarly, two vertices are called ''adjacent'' if they share a common edge (''consecutive'' if the first one is the tail and the second one is the head of an edge), 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.