2. Matrix Representation of Graph

3. CONTENTS
Definition
Types of Matrices
Adjacency Matrix

4. Definition:
A pictorial representation of a graph is very convenient for a visual study. A matrix is a convenient and useful way of representing a graph to a computer. In many applications of graph theory, such as in electrical network analysis and operations research, matrices also turn out to e the natural way of expressing the problem.

5. In this chapter, we study various types of matrices associated with a graph, namely.
Incidence Matrix
Path matrix
Cut-set matrix
Cycle matrix
Circuit Matrix .
Types of Matrices:

6. The matrix element aij is defined as;
Incidence Matrix:
Defn : let G be a graph with n-vertices and e - edges and no self loop, defines on n by e matrix A=[aij] whose n rows corresponds to the n vertices and e columns corresponds to e edges in called incidence matrix.
The matrix element aij is defined as;
Aij = 1 if jth edge ej is incident on the ith vertex vi
0, otherwise
 Incidence matrix of graph G is denoted as A [G]. F

7. Path Matrix:
Let us consider a graph G and let us also consider a vertex pair (a, b) = [Pij] for (a, b) is defined as
Pij = 1 if jth edge in the ith path.
0 otherwise.
Ex. Find the path matrix between vertices v1 & v4
The different paths are
(a, g) (a, f, d) (e, f, g) (e, d)
Then P (v1 v4) =

8. Adjacency Matrix:
Alternative to incidence matrix, it is sometimes more convenient to represent a graph by adjacency matrix or connection matrix. The adjacency matrix of graph G with n vertex and no parallel edges is an n by n symmetric binary matrix X=[Xij].
X ij = 1 if there is an edge between ith and jth
0 if there is no edge between them.

9. ‘0’ diagram means no self loop

10. Cycle Matrix:
The cycle matrix C=[Cij]of g has a row for each cycle and a column for each line with Cij =1 if the ith cycle contains lines xj and Cij =0 otherwise. In contrast to the adjacency and incidence matrices, the cycle matrix does not determine a graph up to isomorphism.
The graph G1 has four different cycle Z1 = {e1, e2},
Z2 = {e3, e5, e7}, Z3= {e4, e6, e7,} & Z4 = {e3, e4, e5, e6}
The cycle matrix is

11. Circuit Matrix:
Let no. of different circuit in a graph G be q and the number of edges in g be c.
Then the circuit matrix is denoted and defined by.
B=[Bij] = 1 if ith circuit includes jth edges
0 otherwise.

12. Ex: Draw the circuit matrix for the following figure.
Soln: Graph has four different circuits;
C1 = {e1, e2} F
C2 = {e3, e5, e7}
C3= {e4, e6, e7,}
C4 = {e3, e4, e5, e6}