Discrete Mathematics with Graph Theory (3rd Edition) 279

# Discrete Mathematics with Graph Theory (3rd Edition) 279 -...

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Section 10.3 277 6. (a) [BB] The (i, j) entry in A 2 is the number of walks of length 2 from i to j. Hence, the sum of all such entries is the total number of walks of length 2. 8. (b) Given a vertex Vi of degree d i , there are d i ways of choosing an edge incident with Vi and d i ways of choosing a second (not necessarily distinct) edge incident with Vi. These two edges, in the order chosen, define a walk of length 2 "through" Vi. Conversely, any walk of length 2 will be chosen in this manner for some Vi. Hence, the total number of walks of length 2 equals E d~, the total number of ways of making such a choice. [ ~ 1 0 n Interchanging vertices 1 and 2 in graph 91 gives graphs 92, so P = 0 0 0 1 0 0 [010111 [ ~ 1 0 0 n 1 0 1 0 0 0 0 1
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Unformatted text preview: (~~B]Al= 0 1 0 0 1 , A 2 = 0 0 1 10000 1 1 0 1 0 1 0 0 0 0 1 (b) [BB] The function cp is an isomorphism because, if the vertices of 91 are relabeled, Vi being replaced by CP(Vi) = Ui, then the adjacency matrix of 91 relative to the Ui'S is A 2. (See Theo-rem 10.3.3.) (c) ~Bl p ~ [~ ~ ~ ~ l 0 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 0 0 9. (a) Al = 0 1 0 1 0 0 , A 2 = 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 (b) The function cP is an isomorphism because, if the vertices of 91 are relabeled, Vi being replaced by cp( Vi) = ui, then the adjacency matrix of 91 relative to the Ui'S is A 2 . (See Theorem 10.3.3.) 0 1 0 0 0 0 0 0 0 0 0 1 (c) P = 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0...
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