Matrix representation of graphs- Adjacency , , Circuit matrix, Fundamental Circuit matrix and , Cut set matrix, matrix .Graph is a set of edges and vertices.

.Graph can be represented in the form of matrix.

.Different matrix that can be formed are:

1. Incidence Matrix

2.

3. Cut-Set Matrix

4. Circuit Matrix

5. Path Matrix .Edge connected to the is known as incidence edge to that vertex.

.If vertex is incident on vertex then put 1 else 0. Vertex Edges 1 a, b 2 a, b, c, f

aij =1, if edge ej is incident on vertex vi 3 c, d, g =0, otherwise 4 d, e 5 d, e, f, g, h 6 h V1 a V6 V2f V5 h b Edges a b c d e f g h V1 1 1 0 0 0 0 0 0 V2 11 1 0 0 1 0 0 c e V3 0 0 1 1 1 0 1 0

Vertex V4 0 0 0 1 1 0 1 0 V50 0 0 1 1 1 1 1 V6 0 0 0 0 0 0 0 1

V3 d V4 . If two vertices are connected by single path than they are known as adjacent vertices.

. If vertex is connected to itself then vertex is said to be adjacent to itself.

.If vertex is adjacent then put 1 else 0.

V1 a V6 V2 f V5 h Vertices b V1 V2 V3V 4 V5 V6

V1 0 1 0 0 0 0 V2 10 1 0 1 0 c e V3 0 1 0 1 1 0 V4 0 0 1 1 1 0

Vertices V5 0 1 1 1 0 1 V6 0 0 0 0 0 1

V3 d V4 .Cut set is a “Set of edges in a graph whose removal leaves the graph disconnected”.

.If edge of graph is a part of given cut set then put 1 else 0. Cut Set Edges th Cij =1, if j cutset contains edge 1 f, g, d =0, otherwise 2 c, g, e 3 h V1 a V6 4 a, b V2f V5 h b Edges ab c d e f g h

1 0 0 0 1 0 1 1 0 c e 2 00 1 0 1 0 1 0 3 0 0 0 0 0 0 0 1 u Sets Cut 4 1 1 0 0 0 0 0 0

V3 d V4 .Circuit can be defined as “A close walk in which no vertex/edge can appear twice”.

.If edge of graph is a part of given circuit then put 1 else 0. Circuit Edges Cij =1, if circuit contains edge 1 d, e, g =0, otherwise 2 c, f, g 3 c, d, e, f V1 a V6 4 a, b V2f V5 h b Edges ab c d e f g h

1 0 0 0 1 1 0 1 0 c e 2 00 1 0 0 1 1 0 3 0 0 1 1 1 1 0 0 Circuits 4 1 1 0 0 0 0 0 0

V3 d V4 .Path can be defined as “A open walk in which no vertex/edge can appear twice”. Path .If edge of graph is a part of given path then put 1 else 0. Edges 1 a, f, h Ex: P(V V ) 2 a, c, g, h P( Vj,Vi)=1,if edge is on path 1, 6 =0, otherwise 3 a, c, d, e, h 4 b, f, h V1 a V6 5 b, c, g, h 6 b,c, d, e, h V2f V5 h Edges b a b c d e f g h

1 1 0 0 0 0 1 0 1 10 1 0 0 0 c e 2 1 1 3 1 0 1 1 1 0 0 1 Paths 4 0 1 0 0 0 1 0 1 50 1 1 0 0 0 1 1 6 0 1 1 1 1 0 0 1 V3 d V4

 In a cycle matrix, if we take only those rows that correspond to a set of fundamental cycles and remove all other rows, we do not lose any information.  The removed rows can be formed from the rows corresponding to the set of fundamental cycles.  For example, in the cycle matrix of the graph given in Figure 10.6, the fourth row is simply the mod 2 sum of the second and the third rows. Fundamental cycles are