Discrete Mathematics with Graph Theory (3rd Edition) 281

Discrete Mathematics with Graph Theory (3rd Edition) 281 -...

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Section 10.3 279 vertices of degree 1 while 91 has only one, the graphs are not isomorphic, so no such P exists, by Theorem 10.3.4. (b) The matrices are the adjacency matrices of the graphs 91,92, respectively, shown to the right. The mapping 'P: 91 ---+ 92 defined by VI ~ U4, V2 ~ U3, V3 ~ Ul, V4 ~ U2, V5 ~ U5 is an isomorphism. Let P be thle oco~esp~nd~ng t e j r - U2t=U4 U5 00010 mutation matrix, that is, P = 0 1 0 0 0 . 10000 U U 92 o 0 0 0 1 1 3 Then P A 1 P T = A 2 . is not unique. Another isomorphism is Md~~vesP= I ~ ~ ~ ~ n This solution (c) The matrices are, respectively, the adjacency matrices of the graphs 91, 92 shown in Exercise 8 of Section 9.3, which we reproduce here. The graphs are not isomorphic (92 on the right has triangles), so no such P exists.
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Unformatted text preview: B~F C~E D 14. [BB] The ith entry on the diagonal of A37 is the number of walks of length 37 from Vi to itself. But in a bipartite graph, you can only get from Vi back to itself in an even number of steps. Hence, the entry is O. 15. (a) [BB] A2 is an adjacency matrix ~ A is the 0 matrix. Proof. For A2 to be an adjacency matrix, it must have all diagonal entries equal to O. But the ith diagonal entry of A2 is the number of walks of length 2 from Vi to itself. Now, if ViVj is an edge of 9, then ViVjVi is a walk of length 2 from Vi to itself, and the ith diagonal entry would not be O. We conclude that 9 cannot have any edges; that is, A is the zero matrix. On the other hand, if A is the zero matrix, certainly A2 = A is an adjacency matrix....
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