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
n

1 edges, but this
occurs
if
and only
if
the sum
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 Summer '10
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 Graph Theory, Planar graph, vertices

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