This preview shows pages 1–4. Sign up to view the full content.
3. REPRESENTING GRAPHS AND GRAPH ISOMORPHISM
195
3. Representing Graphs and Graph Isomorphism
3.1. Adjacency Matrix.
Definition
3.1.1
.
The
adjacency matrix
,
A
= [
a
ij
]
, for a simple graph
G
=
(
V,E
)
, where
V
=
{
v
1
,v
2
,...,v
n
}
, is deﬁned by
a
ij
=
(
1
if
{
v
i
,v
j
}
is an edge of
G,
0
otherwise.
Discussion
We introduce some alternate representations, which are extensions of connection
matrices we have seen before, and learn to use them to help identify isomorphic
graphs.
Remarks
Here are some properties of the adjacency matrix of an undirected graph.
1. The adjacency matrix is always symmetric.
2. The vertices must be ordered: and the adjacency matrix depends on the order
chosen.
3. An adjacency matrix can be deﬁned for multigraphs by deﬁning
a
ij
to be the
number
of edges between vertices
i
and
j
.
4. If there is a natural order on the set of vertices we will use that order unless
otherwise indicated.
3.2. Example 3.2.1.
Example
3.2.1
.
An adjacency matrix for the graph
v
1
v
2
v
3
v
4
v
5
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 3. REPRESENTING GRAPHS AND GRAPH ISOMORPHISM
196
is the matrix
0 1 0 1 1
1 0 1 1 1
0 1 0 1 0
1 1 1 0 1
1 1 0 1 0
Discussion
To ﬁnd this matrix we may use a table as follows. First we set up a table labeling
the rows and columns with the vertices.
v
1
v
2
v
3
v
4
v
5
v
1
v
2
v
3
v
4
v
5
Since there are edges from
v
1
to
v
2
,
v
4
, and
v
5
, but no edge between
v
1
and itself
or
v
3
, we ﬁll in the ﬁrst row and column as follows.
v
1
v
2
v
3
v
4
v
5
v
1
0
1
0
1
1
v
2
1
v
3
0
v
4
1
v
5
1
We continue in this manner to ﬁll the table with 0’s and 1’s. The matrix may
then be read straight from the table.
Example
3.2.2
.
The adjacency matrix for the graph
197
1
u
2
u
3
u
4
u
5
u
is the matrix
M
=
0 0 1 1 1
0 0 1 0 1
1 1 0 1 1
1 0 1 0 1
1 1 1 1 0
3.3. Incidence Matrices.
Definition
3.3.1
.
The
incidence matrix
,
A
= [
a
ij
]
, for the undirected graph
G
= (
V,E
)
is deﬁned by
a
ij
=
(
1
if edge
j
is incident with vertex
i
0
otherwise.
Discussion
Remarks
:
(1) This method requires the edges and vertices to be labeled and depends on
the order in which they are written.
(2) Every column will have exactly two 1’s.
(3) As with adjacency matrices, if there is a natural order for the vertices and
edges that order will be used unless otherwise speciﬁed.
Example
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 11/27/2011 for the course MCH 108 taught by Professor Penelopekirby during the Fall '08 term at FSU.
 Fall '08
 PenelopeKirby

Click to edit the document details