# Graph - Graphs Lecture 9 Discrete Mathematical Structures A...

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Graphs Lecture 9 Discrete Mathematical Structures

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C D A B
Problem of “Crossing River” Problem: A person(P), a wolf(W), a lamb(L) and a cabbage(C) will cross a river by a boat which can carry any two of them once. Wolf and lamb, or, lamb and cabbage, cannot stay together without the person present. Remember that only the person can run the boat.

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Seven Bridges at Königsberg Abstraction Vertices representing objects - areas Edges representing the relationship between objects – connected by a bridge C D A B A C B D
Graph and Diagram Graph G is a triple: G = G V G , E G , ϕ V G and E G are sets G satisgying V G E G = φ , ϕ :E G {{v i , v j }| v i , v j V G } Note: {v i , v j }={v j , v i } A graph can be represented conveniently by some diagram: : each element of V G as a dot, the vertex, and each element of E G as a line segment, the edge, between two vertices. So, V G is called the set of vertices, and E G , the set of

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Relations in Graph Relation defined from edge set to the Cartesian product of vertex set Incidence Relation defined on the set of vertices Adjacency Considering the relations above, a graph can be represented by matrix.
Simple Graph Multiple edges and ring If ϕ is not a injection, that is, exist e i , e j E G , e i e j , but ϕ (e i )= ϕ (e j ), then e i , e j G are called multiple edges . G For any e i E G , if ϕ (e i )={v i , v i }={v i }, then e i is called a ring . Simple graph A graph without ring and multiple edges is called a simple graph .

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Degree of Vertex Degree of vertex d G (v) = number of edge incident to v d G (v) must be a nonnegative integer. Numerical characteristics of the degree of vertex Sum of degree of all vertices in a graph is even. m is the number of edges in the graph. The number of vertices with odd degree must be even. m v d n i i 2 ) ( 1 = =
Complete Graph A graph is a complete graph if and only if any two of its vertices are adjacent. In the sense of isomorphism, for a given n , there is exactly one complete graph, denoted as K n , where n is the number of vertices in the graph. In K

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## This note was uploaded on 03/31/2010 for the course SE C0229 taught by Professor Tao during the Spring '08 term at Nanjing University.

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Graph - Graphs Lecture 9 Discrete Mathematical Structures A...

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