{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Discrete Mathematics with Graph Theory (3rd Edition) 334

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

This preview shows page 1. Sign up to view the full content.

332 Solutions to Exercises 3 -1 0 0 0 -1 -1 2 -1 0 0 0 The value of the (1,1) cofactor is 0 -1 3 -1 0 -1 = 65. 0 0 -1 2 -1 0 0 0 0 -1 3 -1 -1 0 -1 -1 -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, = 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 A BA BA BA BA BA BA BA B NSZV1~/1k:~ C DC DC DC DC DC DC DC D 6. lC 7 (labeled) has 7 5 spanning trees. (See Theorem 12.2.3.) 7. (a) [BB] lC 2,2 has four spanning trees (obtained by deleting each edge in succession). They are all isomorphic to 0 0 0 0 . (b) lC 2,3 has 12 spanning trees as shown.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}