Question

Solve all parts please

5. In the following problems, recall that the adjacency matrix (or incidence matrix) for a simple graph with n vertices is an n x n matrix with entries that are all 0 or 1. The entries on the diagonal are all 0, and the entry in the ih row and jth column is 1 if there is an edge between vertex i and vertex j and is 0 if there is not an edge between vertex i and vertex j. (a) Write down the adjacency matrix for the following simple graph with the vertices ordered according to the labels in the pictures 3 4 5 1 2 7 6 (b) Let A be the adjacency matrix from the previous example. Calculate the matrix A2 using matrix multiplication. Check that in this example the (i,j) entry of the matrix A2 is exactly the number of length 2 walks from vertex i to vertex j (c) Prove that for any simple graph with adjacency matrix A, the (i, j) entry of the matrix A2 gives the number of walks of length 2 from i to j. (d) Challenge: Prove by induction that the (i, j) entry of Ak gives the number of length k walks from vertex i to vertex j. Corollary: A simple graph with n vertices is connected if and only if every entry except the diagonal is positive in the matrix AA2A+ ... +A"-1.

Answer 1

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