Discrete Mathematics with Graph Theory (3rd Edition) 333

Discrete Mathematics with Graph Theory (3rd Edition) 333 -...

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Section 12.2 331 2 -1 0 -1 0 0 -1 3 -1 0 0 -1 Kirchhoff's matrix is 0 -1 3 -1 0 -1 -1 0 -1 3 -1 0 0 0 0 -1 2 -1 0 -1 -1 0 -1 3 3 -1 0 0 -1 -1 3 -1 0 -1 The (1,1) cofactor is 0 -1 3 -1 0 0 0 -1 2 -1 -1 -1 0 -1 3 3 -1 0 -1 -1 0 0 -1 -1 0 0 -1 =3 -1 3 -1 0 -1 3 -1 0 3 -1 0 -1 0 -1 2 -1 + 0 -1 2 -1 -1 3 -1 0 -1 0 -1 3 -1 0 -1 3 0 -1 2 -1 ~3 {3 3 -1 0 -1 0 -1 -1 0 -1 } -1 2 -1 + -1 2 -1 + 3 -1 0 0 -1 3 0 -1 3 -1 2 -1 3 -1 0 -1 3 -1 + (-1) -1 2 -1 - (-1) 0 -1 2 0 -1 3 -1 0 -1 - { (-1) -1 0 -1 3 -1 0 }
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Unformatted text preview: 3 -1 0 - (-1) -1 3 -1 -1 2 -1 0 -1 2 = 3{3[3(5) + (-3)] + [( -1)5 + (-1)] + [( -1)1 + -5]} -[3(5) + (-3)] + [(-1)1 -5]- {( -1)[-1(1) - 5] + 3(5) + (-2)} = 3(24) - 12 - 6 - 19 = 35. We conclude that there are 35 spanning trees. 5 (d) Here is one possible answer. The first and 'm' m d second trees are isomorphic, but neither is isomorphic to the third. 1 2 3 2 -1 0 0 0 -1 0 -1 3 -1 0 0 0 -1 0 -1 2 -1 0 0 0 Kirchhoff's matrix is 0 0 -1 3 -1 0 -1 0 0 0 -1 3 -1 -1 -1 0 0 0 -1 3 -1 0 -1 0 -1 -1 -1 4...
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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.

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