Discrete Mathematics with Graph Theory (3rd Edition) 312

# Discrete Mathematics with Graph Theory (3rd Edition) 312 -...

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310 Solutions to Exercises (b) See Exercise 18(b) of Section 11.2. The score sequences, left to right, are 3,2,1,0, 3,1,1,1, 2,2,2,0 and 2, 2, 1, 1. Only the leftmost tournament is transitive. 2. [BB] Since for each pair of (distinct) vertices Vi, Vj precisely one of ViVj, VjVi is an arc, in the adja- cency matrix A, for each i =I j, precisely one of aij, aji is Thus, A + AT has o's on the diagonal and l's in every off-diagonal position. 3. (a) If A is the adjacency matrix of a tournament, then A + has o's on the diagonal and l's everywhere else, by Exercise 2. This matrix is J - I. On the other hand, if A + = J - I for some adjacency matrix then 2aii = 0, so each diagonal entry of A is O. Also, if i aij + aji = 1 with each either 0 or 1 says that exactly one of these entries is 1, while the other is O. Thus A is the adjacency matrix of a tournament. (b) Let A be the adjacency matrix of a tournament and let B = AT. Then B + BT = + so B is also the adjacency matrix of a tournament, by Exercise 3(a) (c) Using Exercise 3(a), it suffices to show that P ApT = J - We know A = J - I, so = p J pT _
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