Discrete Mathematics with Graph Theory (3rd Edition) 335

Discrete Mathematics with Graph Theory (3rd Edition) 335 -...

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Section 12.2 333 When k = 1, there are n spanning trees. Thus, the total number of spanning trees is 9. [BB] The edge in question is a bridge (see Definition 12.6.2); that is, its removal disconnects the graph. To see why, call the edge e. If 9 " {e} were connected, it would have a spanning tree. However, since 9 " {e} contains all the vertices of g, any spanning tree for it is also a spanning tree for g. We have a contradiction. 10. The only spanning tree of 9 is 9 itself. Any spanning tree different from 9 would have to contain some edge not in 9 and there are no such edges. 11. The addition of any edge to a tree produces a circuit. We are told that if 7 is a spanning tree, the addition of e does not produce a circuit, so e must already belong to 7. 12. (a) [BB] Say the edge is e and 7 is any spanning tree. If e is not in 7, then 7 U {e} must contain a circuit. Deleting any edge of this circuit other than e gives another spanning tree which includes e.
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