Discrete Mathematics with Graph Theory (3rd Edition) 332

# Discrete Mathematics with Graph Theory (3rd Edition) 332 -...

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330 Solutions to Exercises 2 -1 -1 0 0 -1 3 -1 0 -1 Kirchhoff's matrix is -1 -1 3 -1 0 . The value of the (5, 1) cofactor is 0 0 -1 2 -1 0 -1 0 -1 2 -1 -1 0 0 -1 0 -1 3 0 -1 3 -1 0 -1 3 -1 0 -1 -1 0 -1 3 -1 0 + 0 -1 2 -1 -1 2 -1 0 2 -1 = -[(-1)(1) + (-1)(5)] + [3(1) - 1( -2)] = II. We conclude that there are 11 spanning trees. 3 4 (b) Here is one possible answer. The first and second 2{)5\V W trees are isomorphic, but neither is isomorphic to the 1 6 third. 2 -1 0 -1 0 0 -1 2 -1 0 0 0 Kirchhoff's matrix is 0 -1 3 -1 0 -1
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Unformatted text preview: -1 0 -1 3 -1 0 0 0 0 -1 2 -1 0 0 -1 0 -1 2 2 -1 0 0 0 -1 3 -1 0 -1 The (1,1) cofactor is 0 -1 3 -1 0 0 0 -1 2 -1 0 -1 0 -1 2 ~2{3 3 -1 0 -1 -1 0 -1 3 -1 } -1 2 -1 + 0 2 -1 + 0 -1 2 0 -1 2 -1 -1 2 -1 0 -1 3 -1 0 + (-1) -1 2 -1 0 -1 2 = 2{3[3(3) + 1( -2)] + (-1)3 + 1( -1) + (-1)(1) + (-1)(5)} + (-1)[3(3) + 1( -2)] = 2(11) + (-1)(7) = 15. We conclude that there are 15 spanning trees. (c) Here is one possible answer. The first and second trees are isomorphic, but neither is isomorphic to the third. 1 2 2IHK 4 3...
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