Glossarygraph theory
A loop ingraph ordigraphan edge eE whose endpoints aresame vertex. A digraph or graphcalled simple if thereno loopsthereat most one edge between any pairvertices.
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An edge connects two vertices; these two verticessaidbe incident toedge. The valency (or degree) ofvertex isnumberedges incidentit,loops being counted twice. Inexample graph vertices 13 havevalency2, vertices 2,45 havevalency3vertex 6 hasvalency1. If Efinite, thentotal valency ofverticesequaltwicenumberedges. Indigraph, we distinguishout degree (=the numberedges leavingvertex) andin degree (=the numberedges enteringvertex). The degree ofvertexequal tosum ofout degree andin degree.
Two verticesconsidered adjacent if an edge exists between them. Inabove graph, vertices 12adjacent, but vertices 24not. The setneighbors forvertex consistsall vertices adjacentit. Inexample graph, vertex 1 has two neighbors: vertex 2node 5. Forsimple graph,numberneighbors thatvertex has coincidesits valency.
In computers,finite directed or undirected graph (with n vertices, say)often represented by its adjacency matrix: an n-by-n matrix whose entryrow icolumn j givesnumberedges fromi-th toj-th vertex.
A path issequencevertices such that from eachits vertices therean edge tosuccessor vertex. A pathconsidered simple if none ofvertices inpathrepeated. Two pathsindependent ifdo not have any vertexcommon, exceptfirstlast one.
The length ofpath isnumberedges thatpath uses, counting multiple edges multiple times. Inexample graph, (1, 2, 5, 1, 2, 3) ispathlength 5,(5, 2, 1) issimple pathlength 2.
A weighted graph associatesvalue (weight)every edge ingraph. The weightpath inweighted graph issum ofweights oftraversed edges. Sometimesword costused insteadweight.
If itpossibleestablishpath from any vertexany other vertex ofgraph,graphsaidbe connected. If italways possibleestablishpath from any vertexany other vertex even after removing k-1 vertices, thengraphsaidbe k-connected. Note thatgraphk-connected ifonly ifcontains k independent paths between any two vertices. The example graph aboveconnected (and therefore 1-connected), but not 2-connected.
A cycle (or circuit) ispath that beginsends withsame vertex. Cycleslength 1loops. Inexample graph, (1, 2, 3, 4, 5, 2, 1) iscyclelength 6. A simple cycle iscycle which has length at least 3in whichbeginning vertex only appears once more, asending vertex, andother vertices appear only once. Inabove graph (1, 5, 2, 1) issimple cycle. A graphcalled acyclic ifcontains no simple cycles.
An articulation point isvertex whose removal disconnectsgraph. A bridgean edge whose removal disconnectsgraph. A biconnected component ismaximal setedges such that any two edges inset lie oncommon simple cycle. The girth ofgraph islength ofshortest simple cycle ingraph. The girthan acyclic graphdefinedbe infinity.
A tree isconnected acyclic simple graph. Sometimes, one vertex oftreedistinguished,calledroot. Treescommonly used as data structurescomputer science (see tree data structure).
A forest issettrees; equivalently,forestany acyclic graph.
A subgraph ofgraph G isgraph whose vertex set issubset ofvertex setG, whose edge set issubset ofedge setG,such thatmap w isrestriction ofmap from G.
A spanning subgraph ofgraph G issubgraph withsame vertex set as G. A spanning tree isspanning subgraph that istree. Every graph hasspanning tree.
A complete graph issimple graphwhich every vertexadjacentevery other vertex. The example graphnot complete. The complete graph on n verticesoften denoted by Kn. It has n(n-1)/2 edges (correspondingall possible choicespairsvertices).
A regular graph has all vertices ofsame valency.
A universal graph inclass Kgraphs issimple graphwhich every elementK can be embbeded assubgraph.
A planar graphone which can be drawn inplane without any two edges intersecting. The example graphplanar;complete graph on n vertices,n> 4,not planar.
An Eulerian path ingraph ispath that uses each edge precisely once. If suchpath exists,graphcalled traversable. An Eulerian cycle iscycleuses each edge precisely once.
There isdual toEulerian path/cycle concept. A Hamiltonian path ingraph ispath that visits each vertex onceonly once; andHamiltonian cycle iscycle which visits each vertex onceonly once.
The example graph does not contain an Eulerian path, butdoes containHamiltonian path.
An empty graph isgraph whose edge setempty.
The null graph isgraph whose edge setvertex setempty.
An independent set ingraph issetpairwise nonadjacent vertices. Inexample above, vertices 1,3,6 form an independent set3,5,6another independent set.
A clique (pronounced "click") ingraph issetpairwise adjacent vertices. Inexample graph above, vertices 1, 25 formclique.
A bipartite graphany graph whose vertices can be divided into two sets, such that thereno edges between vertices ofsame set. A graph can be proved bipartite if there do not exist any circuitsodd length.
A k-partite graph or k-colorable graph isgraph whose vertices can be partitioned into k disjoint subsets such that thereno edges between vertices insame subset. A 2-partite graph issame asbipartite graph.
A tournamentdirected graphwhich each pairverticesconnected by exactly one arc.
See also: Graph (mathematics), Graph theory, Listgraph theory topics
