2. B D A E F H a) Create Adjacency Matrix of the graph. b) Find the Strongly Connected Components.

2. (15 pts.) the adjacency matrix of G. Let n be the number of vertices so that the matrix M is n x n matrix. For any matrix A, let us denote the element of i-th row and j-th column of the matrix A by A[i, j]. We are given a graph G = (V, E); G could be a directed graph or undirected graph. Let M be 1. Consider the square of the adjacency matrix M. For all i and j, show that M²[i, j] is the number of different paths of length 2 from the i-th vertex to the j-th vertex. It should be explained or proved as clearly as possible. 2. For any positive integer k, show that Mk[i, j] is the number of different paths vertex to the j-th vertex. You may use induction on k to prove it. length k from the i-th 3. Assume that we are given a positive integer k. Design an algorithm to find the number of different paths of length k from the i-th vertex to j-th vertex for all pairs of (i, j). The time complexity of your algorithm should be O(n³ log k). You can get partial credits if you design an algorithm…

(a) Let G be a simple undirected graph with 18 vertices and 53 edges such that the degreesof G are only 3 and 7. Suppose there are a vertices of degree 7 and b vertices ofdegree 3. Find a and b. To receive any credit for this problem you must write completesentences, explain all of your work, and not leave out any details. problem 1, continued(b) Recall that a graph G is said to be k-regular if and only if every vertex in G has degreek. Draw all 3-regular simple graphs with 12 edges (mutually non-isomorphic). Hint:there are six of them. To receive credit for this problem, you should explain, as well aspossible, why all of your graphs are mutually non-isomorphic.

4. Using Floyd's algorithm, compute the distance matrix for the weight directed graph defined by the following matrix: 0 4 -2] 6 2 3 -3 2 0 4 ∞o 5 0 Show the intermediate matrices after each iteration of the outermost loop.