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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 Summer '10
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 Graph Theory, Matrices, Binary relation, adjacency matrix

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