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
from
i
to
j,
and so the
(i,j)
entry
of
Ak
is
1.
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 Summer '10
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 Linear Algebra, Determinant, Graph Theory, Matrices, Characteristic polynomial, Diagonal matrix

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