Draw a graph that has
[
0
0
0
1
2
0
0
0
1
1
0
0
0
2
1
1
1
2
0
0
2
1
1
0
0
]
as its adjacency matrix. Is this graph biparitite?
Definition: Given an
m
×
n
matrix A whose ijth entry is denoted
a
i
j
, the transpose of A is the matrix A’ whose ijth entry is
a
i
j
, for each
i
=
1
,
2
,
…
,
m
and
j
=
1
,
2
,
…
,
n
.
Note that the first row of A becomes the first column of
A
t
, the second row of A becomes the second column of
A
t
, and so forth. For instance.
if
A
=
[
0
2
1
1
2
3
]
, then
A
t
=
[
0
1
2
2
1
3
]
.
b. Show that a graph with ii vertices is bipartite if, and only if, for some labeling of its vertices, its adjacency matrix has the form
[
O
A
A
t
O
]
where A is a
k
×
(
n
−
k
)
matrix for some integer k such that
0
<
k
<
n
, the top left O represents a
k
×
k
matrix all of whose entries are 0,
A
t
is the transpose of A, and the bottom right O represents an
(
n
−
k
)
×
(
n
−
k
)
matrix all of whose entries are 0.