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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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 Summer '10
 any
 Graph Theory

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