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MATH 245 CH17 - DISCRETE DISCRETE MATHEMATICS W W L CHEN c...

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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 17 GRAPHS 17.1. Introduction A graph is simply a collection of vertices, together with some edges joining some of these vertices. Example 17.1.1. The graph 1 2 3 4 5 has vertices 1 , 2 , 3 , 4 , 5, while the edges may be described by { 1 , 2 } , { 1 , 3 } , { 4 , 5 } . In particular, any edge can be described as a 2-subset of the set of all vertices; in other words, a subset of 2 elements of the set of all vertices. Definition. A graph is an object G = ( V, E ), where V is a finite set and E is a collection of 2-subsets of V . The elements of V are known as vertices and the elements of E are known as edges. Two vertices x, y V are said to be adjacent if { x, y } ∈ E ; in other words, if x and y are joined by an edge. Example 17.1.2. In our earlier example, V = { 1 , 2 , 3 , 4 , 5 } and E = {{ 1 , 2 } , { 1 , 3 } , { 4 , 5 }} . The vertices 2 and 3 are both adjacent to the vertex 1, but are not adjacent to each other since { 2 , 3 } 6∈ E . We can also represent this graph by an adjacency list 1 2 3 4 5 2 1 1 5 4 3 where each vertex heads a list of those vertices adjacent to it. Chapter 17 : Graphs page 1 of 11
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Discrete Mathematics c W W L Chen, 1992, 2008 Remark. Note that our definition does not permit any edge to join the same vertex, so that there are no “loops”. Note also that the edges do not have directions. Example 17.1.3. For every n N , the wheel graph W n = ( V, E ), where V = { 0 , 1 , 2 , . . . , n } and E = {{ 0 , 1 } , { 0 , 2 } , . . . , { 0 , n } , { 1 , 2 } , { 2 , 3 } , . . . , { n - 1 , n } , { n, 1 }} . We can represent this graph by the adjacency list below. 0 1 2 3 . . . n - 1 n 1 0 0 0 . . . 0 0 . . . 2 3 4 . . . n 1 n n 1 2 . . . n - 2 n - 1 For example, W 4 can be illustrated by the picture below. 17–2 W W L Chen : Discrete Mathematics Remark. Note that our definition does not permit any edge to join the same vertex, so that there are no “loops”. Note also that the edges do not have directions. Example 17.1.3. For every n N , the wheel graph W n = ( V, E ), where V = { 0 , 1 , 2 , . . . , n } and E = {{ 0 , 1 } , { 0 , 2 } , . . . , { 0 , n } , { 1 , 2 } , { 2 , 3 } , . . . , { n - 1 , n } , { n, 1 }} . We can represent this graph by the adjacency list below. 0 1 2 3 . . . n - 1 n 1 0 0 0 . . . 0 0 . . . 2 3 4 . . . n 1 n n 1 2 . . . n - 2 n - 1 can be illustrated by the picture below. 1 4 0 2 3 Example 17.1.4. For every n N , the complete graph K n = ( V, E ), where V = { 1 , 2 , . . . , n } and E = {{ i, j } : 1 i < j n } . In this graph, every pair of distinct vertices are adjacent. For example, K 4 can be illustrated by the picture below. 1 2 4 3 In Example 17.1.4, we called K n the complete graph. This calls into question the situation when we may have another graph with n vertices and where every pair of dictinct vertices are adjacent. We have then to accept that the two graphs are essentially the same.
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