Discrete Mathematics with Graph Theory (3rd Edition) 334

Discrete Mathematics with Graph Theory (3rd Edition) 334 -...

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332 Solutions to Exercises 3 -1 0 0 0 -1 2 -1 0 0 0 The value of the (1,1) cofactor is 0 -1 3 0 = 65. 0 0 0 0 0 0 0 -1 4 We conclude that there are 65 spanning trees. 3. lC n has nn-2 spanning trees each with n - 1 edges. Hence, the total number of all edges used in all spanning trees is (n - l)nn-2. Now each of the (~) edges in lC n is equally likely to be included in a spanning tree. Hence, the number of spanning trees containing e is (n - (i)nn-2 = 2nn-3. 4. [BB] By Theorem 12.2.3, the numbers are 1- 1 = 1, 2° = 1, 3 1 = 3, 4 2 = 16, 53 = 125, and 6 4 = 1296. 5. [BB] A BA BA BA BA BA BA BA B :J~nXClXUX C DC DC DC DC DC DC DC D NSZV1~/1k:~ 6. lC 7 (labeled) has 7 5 spanning trees. (See Theorem 12.2.3.) 7. (a) [BB] 2,2 has four spanning trees (obtained by deleting each edge in succession). They are all isomorphic to 0 0 0 0 . (b) 2,3 has 12 spanning trees as shown. The graphs in the top row form one isomorphism class and those in the bottom row a second.
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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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