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s6_3 - 3 REPRESENTING GRAPHS AND GRAPH ISOMORPHISM 195 3...

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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
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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
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3. REPRESENTING GRAPHS AND GRAPH ISOMORPHISM 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 3.3.1 . The incidence matrix for the graph
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3. REPRESENTING GRAPHS AND GRAPH ISOMORPHISM 198 v 1 v 2 v 3 v 4 v 5 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 is the matrix 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 0 Again you can use a table to get the matrix. List all the vertices as the labels for the rows and all the edges for the labels of the columns.
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