Section 12.1 327 (b) [BB] Since each tree with five vertices has all vertices of degree at most four, there is one isomer for each such tree, the C atoms corresponding to the vertices. There are three isomers of C5H12• 14. (a) 0 0 0 0 0 0 * (b) There is one isomer for each tree (with six vertices) in which all vertices have degree at most four, the C atoms corresponding to the vertices. There are five isomers of C6H14. 15. [BB] A beta index less than 1 says that there are fewer edges than vertices. One possibility is that the graph is not connected; in other words, there exist two cities such that it is impossible to fly from one city to the other. If the graph is connected, then it must be a tree by Theorem 12.1.6. This means there is a unique way of flying from any city to any other city. 16. By Proposition 9.2.5, the sum of the degrees of the vertices is twice the number of edges. According to Theorem 12.1.6, a connected graph with n vertices is a tree if and only if it has n -1 edges, but this occurs if and only if the sum
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