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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
C
5
H
12•
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
C
6
H
14 .
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
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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.
 Summer '10
 any
 Graph Theory

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