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Adjacency matrices and graphs Figure 3: An undirected graph with n = 5 vertices. For a given undirected graph with no self-loops and at most one edge between any pair of nodes (i.e., a simple graph), as in Figure 3, we associate an n x n matrix A, such that if there is an edge between node i and node j, Aij = 1; otherwise, Aij = 0. This matrix is called the adjacency matrix of the graph. The graph in Figure 3 has an adjacency matrix. 1. (2 points) Prove the following result: for a positive integer k, the matrix has an interesting interpretation: the entry in the row and column gives the number of walks of length k (i.e., a collection of k edges) leading from vertex i to vertex j. Hint: prove this by induction on k, and look at the matrix-matrix product Ak-1A.

          Adjacency matrices and graphs
Figure 3: An undirected graph with n = 5 vertices.
For a given undirected graph with no self-loops and at most one edge between any pair of nodes (i.e., a simple graph), as in Figure 3, we associate an n x n matrix A, such that if there is an edge between node i and node j, Aij = 1; otherwise, Aij = 0.
This matrix is called the adjacency matrix of the graph. The graph in Figure 3 has an adjacency matrix.
1. (2 points) Prove the following result: for a positive integer k, the matrix has an interesting interpretation: the entry in the row and column gives the number of walks of length k (i.e., a collection of k edges) leading from vertex i to vertex j. Hint: prove this by induction on k, and look at the matrix-matrix product Ak-1A.

adjacency matrices and graphs figure 3 an undirected graph with n 5 vertices for a given undirected graph with no self loops and at most one edge between any pair of nodes ie simple graph as 64617

Added by Alexandra H.

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