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单词 Graph
释义

graph


A graph G is an ordered pairMathworldPlanetmath of disjoint sets (V,E) such that E is a subset of the set V(2) of unordered pairs of V. V and E are always assumed to be finite, unless explicitly stated otherwise. The set V is the set of vertices (sometimes called nodes) and E is the set of edges. If G is a graph, then V=V(G) is the vertex set of G, and E=E(G) is the edge set.Typically, V(G) is defined to be nonempty. If x is a vertex of G, we sometimes write xG instead of xV(G).

An edge {x,y} (with xy) is said to join the vertices x and y and is denoted by xy. One says that the edges xy and yx are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath; the vertices x and y are the endvertices of this edge. If xyE(G), then x and y are adjacent, or neighboring, vertices of G, and the vertices x and y are incidentPlanetmathPlanetmath with the edge xy. Two edges are adjacent if they have at least one common endvertex. Also, xy means that the vertex x is adjacent to the vertex y.

Notice that this definition allows pairs of the form {x,x}, which would correspond to a node joining to itself. Some authors explicitly disallow this in their definition of a graph.


Some graphs.

Note: Some authors include multigraphsMathworldPlanetmath in their definition of a graph. In this notation, the above definition corresponds to that of a simple graph. A graph is then simple if there is at most one edge joining each pair of nodes.

Adapted with permission of the author from by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Titlegraph
Canonical nameGraph
Date of creation2013-03-22 11:57:54
Last modified on2013-03-22 11:57:54
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id34
Authormathcam (2727)
Entry typeDefinition
Classificationmsc 05C99
Related topicLoopOfAGraph
Related topicNeighborhoodOfAVertex
Related topicEulersPolyhedronTheorem
Related topicDigraphMathworldPlanetmath
Related topicTree
Related topicSpanningTree
Related topicConnectedGraph
Related topicCycle
Related topicGraphTheory
Related topicGraphTopology
Related topicSubgraphMathworldPlanetmath
Related topicSimplePath
Related topicEulerPath
Related topicDiameter3
Related topicDistanceInAGraph
Related topicGraphHomomorphism
Related topicPseudographMathworldPlanetmath
Related topicMultigraph
Related topicOrderOf
Definesedge
Definesvertex
Definesendvertex
Definesadjacent
Definesincident
Definesjoin
Definesvertices
Definessimple graph
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