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280
Solutions to Exercises
(b)
A
3
is the adjacency matrix of a graph
9
~
the connected components of
9
are of the form
0
or
00.
Proof.
Suppose that
A
3
is the adjacency matrix of a graph
9.
Then all entries of
A3
must be 0 or
1.
If
9
contains any vertex of degree
d
>
1,
say
v,
then we have
wvx.
But now, there are two
walks oflength 3 from
v
to
w;
namely,
vxvw
and
vwvw,
so the
(v, w)
entry of
A3
would be bigger
than
1.
We conclude that every vertex of
9
is of degree 0 or 1; that is, the connected components
of
9
are of the form
0
or
00.
Conversely, if
9
is a graph whose connected components are all of
this form, then all entries of
A3
are 0 or 1
(aij
=
1
~
ViOOVj).
Note that the diagonal entries
are all
0
since there are no triangles. Hence,
A
3
is the adjacency matrix of some graph.
•
16. (+) Suppose the graph is connected. Then there is a walk between every pair of vertices. Let
n
be
the length of the longest such walk. Then for any
i
and
j,
there is a walk of length
k
::;
n
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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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