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MATH 245 CH18

# MATH 245 CH18 - DISCRETE MATHEMATICS DISCRETE DISCRETE...

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DISCRETE MATHEMATICS W W L CHEN c ± W W L Chen, 1992, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for ﬁnancial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 18 WEIGHTED GRAPHS 18.1. Introduction We shall consider the problem of spanning trees when the edges of a connected graph have weights. To do this, we ﬁrst consider weighted graphs. Definition. Suppose that G = ( V,E ) is a graph. Then any function of the type w : E N is called a weight function. The graph G , together with the function w : E N , is called a weighted graph. Example 18.1.1. Consider the connected graph described by the following picture. 123 456 Then the weighted graph DISCRETE MATHEMATICS WWL CHEN c ² WWL Chen, 1992, 2003. This chapter is available free to all individuals, on the understanding that it is not to be used for Fnancial gains, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 18 WEIGHTED GRAPHS 18.1. Introduction We shall consider the problem of spanning trees when the edges of a connected graph have weights. To do this, we Frst consider weighted graphs. Definition. Suppose that G =( V, E )isa graph. Then any function of the type w : E N is called aweight function. The graph G , together with the function w : E N ,is called a weighted graph. Example 18.1.1. Consider the connected graph described by the following picture. Then the weighted graph 3 2 5 1 4 6 7 has weight function w : E N , where E = {{ 1 , 2 } , { 1 , 4 } , { 2 , 3 } , { 2 , 5 } , { 3 , 6 } , { 4 , 5 } , { 5 , 6 }} and w ( { 1 , 2 } )=3 ,w ( { 1 , 4 } )=2 ( { 2 , 3 } )=5 ( { 2 , 5 } )=1 , w ( { 3 , 6 } )=4 ( { 4 , 5 } )=6 ( { 5 , 6 } )=7 . has weight function w : E N , where E = {{ 1 , 2 } , { 1 , 4 } , { 2 , 3 } , { 2 , 5 } , { 3 , 6 } , { 4 , 5 } , { 5 , 6 }} and w ( { 1 , 2 } ) = 3 , w ( { 1 , 4 } ) = 2 , w ( { 2 , 3 } ) = 5 , w ( { 2 , 5 } ) = 1 , w ( { 3 , 6 } ) = 4 , w ( { 4 , 5 } ) = 6 , w ( { 5 , 6 } ) = 7 . Chapter 18 : Weighted Graphs page 1 of 7

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Discrete Mathematics c ± W W L Chen, 1992, 2008 Definition. Suppose that G = ( V,E ), together with a weight function w : E N , forms a weighted graph. Suppose further that G is connected, and that T is a spanning tree of G . Then the value w ( T ) = X e T w ( e ) , the sum of the weights of the edges in T , is called the weight of the spanning tree T . 18.2. Minimal Spanning Tree Clearly, for any spanning tree T of G , we have w ( T ) N . Also, it is clear that there are only ﬁnitely many spanning trees T of G . It follows that there must be one such spanning tree T where the value w ( T ) is smallest among all the spanning trees of G .
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MATH 245 CH18 - DISCRETE MATHEMATICS DISCRETE DISCRETE...

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