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. •...
View
Full
Document
This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory, Matrices

Click to edit the document details