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Presented By:
Abhishek Pachisia
B.Tech-IT(V Sem)
   090102801
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. Adjacency Matrix

   3. Cut-Set Matrix

   4. Circuit Matrix

   5. Path Matrix
Edge connected to the vertex 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
                       V2   f   V5 h
           b                                                            Edges
                                                              a   b    c d e         f    g h
                                                         V1   1   1    0   0   0     0    0   0
                                                         V2   1   1    1   0   0     1    0   0
                                                Vertex

                   c                 e                   V3   0   0    1   1   1     0    1   0
                                                         V4   0   0    0   1   1     0    1   0
                                                         V5   0   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 V3 V V5 V6
                                                                                4
                                                                V1   0 1 0 0 0 0
                                                                V2   1 0 1 0 1 0

                                                     Vertices
                  c                     e                       V3   0 1 0 1 1 0
                                                                V4   0 0 1 1 1 0
                                                                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
  Cij =1, if  jth
               cutset contains edge                               1       f, g, d
      =0, otherwise                                               2       c, g, e
                                                                  3       h
     V1       a                                 V6                4       a, b
                      V2       f   V5       h
          b                                                                      Edges
                                                                      a   b    c d e        f   g   h

                                                                1     0   0    0    1   0   1   1   0
                                                     Cut Sets

                  c                     e                       2     0   0    1    0   1   0   1   0
                                                                3     0   0    0    0   0   0   0   1
                                                                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
                      V2   f   V5 h
          b                                                              Edges
                                                              a   b    c d e           f   g   h

                                                        1     0   0    0       1   1   0   1   0
                                             Circuits

                  c                 e                   2     0   0    1       0   0   1   1   0
                                                        3     0   0    1       1   1   1   0   0
                                                        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”.

If edge of graph is a part of given path then put 1 else 0.             Path       Edges
                                                                          1         a, f, h
  P( Vj,Vi)=1,if edge is on path                        Ex: P ( V1,V6)    2         a, c, g, h
            =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
                 V2   f   V5 h                                Edges
     b
                                                    a   b   c d e        f      g   h

                                                1   1   0   0   0    0   1      0   1
                                                        0
                                        Paths



             c                 e                2   1       1   0    0   0      1   1
                                                3   1   0   1   1    1   0      0   1
                                                4   0   1   0   0    0   1      0   1
                                                5   0   1   1   0    0   0      1   1
                                                6   0   1   1   1    1   0      0   1
         V3           d   V4
Matrix Representation Of Graph

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Matrix Representation Of Graph

  • 2. 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. Adjacency Matrix 3. Cut-Set Matrix 4. Circuit Matrix 5. Path Matrix
  • 3. Edge connected to the vertex 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 V2 f V5 h b Edges a b c d e f g h V1 1 1 0 0 0 0 0 0 V2 1 1 1 0 0 1 0 0 Vertex c e V3 0 0 1 1 1 0 1 0 V4 0 0 0 1 1 0 1 0 V5 0 0 0 1 1 1 1 1 V6 0 0 0 0 0 0 0 1 V3 d V4
  • 4.  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 V3 V V5 V6 4 V1 0 1 0 0 0 0 V2 1 0 1 0 1 0 Vertices c e V3 0 1 0 1 1 0 V4 0 0 1 1 1 0 V5 0 1 1 1 0 1 V6 0 0 0 0 0 1 V3 d V4
  • 5. 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 Cij =1, if jth cutset contains edge 1 f, g, d =0, otherwise 2 c, g, e 3 h V1 a V6 4 a, b V2 f V5 h b Edges a b c d e f g h 1 0 0 0 1 0 1 1 0 Cut Sets c e 2 0 0 1 0 1 0 1 0 3 0 0 0 0 0 0 0 1 4 1 1 0 0 0 0 0 0 V3 d V4
  • 6. 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 V2 f V5 h b Edges a b c d e f g h 1 0 0 0 1 1 0 1 0 Circuits c e 2 0 0 1 0 0 1 1 0 3 0 0 1 1 1 1 0 0 4 1 1 0 0 0 0 0 0 V3 d V4
  • 7. Path can be defined as “A open walk in which no vertex/edge can appear twice”. If edge of graph is a part of given path then put 1 else 0. Path Edges 1 a, f, h P( Vj,Vi)=1,if edge is on path Ex: P ( V1,V6) 2 a, c, g, h =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 V2 f V5 h Edges b a b c d e f g h 1 1 0 0 0 0 1 0 1 0 Paths c e 2 1 1 0 0 0 1 1 3 1 0 1 1 1 0 0 1 4 0 1 0 0 0 1 0 1 5 0 1 1 0 0 0 1 1 6 0 1 1 1 1 0 0 1 V3 d V4