Let G be a connected graph with n ≥ 2 vertices. Let A be the adjacency matrix of G. Prove that the diameter of G is the least number d such that all the non-diagonal entries of the matrix A are positive.
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- 2. Let G be the following graph. V1 V2 V3 V5 VA (a) Let A be the adjacency matrix of G. Find A. (b) Using only A and its powers, determine how many walks of length EXACTLY 3 are there starting at 2 and ending at v4. Explain. (c) Using only A and its powers, determine how many walks of length AT MOST 3 are there starting at 2 and ending at v4. Explain.Let G be a graph with n ≥ 2 vertices x1, x2, . . . , xn, and let A be the adjacency matrixof G. Prove that if G is connected, then every entry in the matrix A^n−1 + A^nis positive.Draw the multigraph G whose adjacency matrix is 0 1 2 0 1 1 1 1 A 2 1 0 0 \o 1 0 1/ Show that the maximum number of edges in a graph with n-vertices is *C2
- For the directed graphs, draw the graph and show: A, A2 , A3, A4, A* , where A is the adjacency matrix for the edge set as a relation. (a) G1 = [{1, 2, 3, 4}]; (b) G2 = [{1, 2, 3, 4}]; Hint: Long matrix multiplications are not required, think about what for each of the matrices it means for a cell to contain a value of 1.Let M be the incidence matrix and A the adjacency matrix of a graph G.(a) Show that every column sum of M is 2.(b) What are the column sums of A?Let A be the adjacency matrix of a complete graph K4.(a) Write down matrix A.(b) How many possible walks with length 2 are there from a (any) node to itself (e.g., from Node 2 to itself) (c) How many possible walks with length 3 are there from a (any) node to the other node (e.g., from Node 2 to Node 3 or Node 1 to Node 4)
- Q2 Let G be the following graph with 5 vertices: 3 (4. 1) Write the adjacency matrix of G 2) Enter your adjacency matrix on Pari (or other software) and use it to calculate the number of paths of length 10 starting at the vertex 1. How many of these paths end at vertex 1? Vertex 2, 3, 4, and 5?Consider a set of webpages hyperlinked by the directed graph given to the right. Find the Google matrix for the graph and compute the PageRank of each page in the set. The Google matrix is G = (Round any values in the matrix to five decimal places as needed.) N 5Consider the relation R = {(a,b), (b, b), (b, c), (c, b), (c, c), (d, a), (d, c), (d, d)} on {a, b, c, d}. (a) Draw the digraph for R. (b) Write the adjacency matrix for R with respect to vertices in alphabetical order. (c) Draw the digraph for R o R.
- In chess. A “knight’s move” consists of two squares either vertically or horizontally and then one square is a perpendicular direction. Depending on where the knight is situated, he has a minimum mobility of two moves—when in a corner—and a maximum mobility of eight moves. Let C be a graph with v=64, its vertices corresponding to the squares of a chessboard. Let two vertices of C be joined by an edge whenever a knight can go from one of the corresponding squares to the other in one move. Does C have an Euler Walk? Explain, but you do not have to draw C to answer.How many 1's are in the adjacency matrix representation of the directed graph {{1,2, 3}, {(1, 2), (2, 3), (3, 1)}} 6.Consider the network with N = 4 nodes and adjacency matrix A given by 0 1 1 1 0 1 1 A = 0000 0000 (a) Calculate the number of links in the network and draw the network. Is the network directed or undirected? (b) Is the network weakly connected? Is the network strongly connected? List the nodes belonging to each of the strongly-connected components in the network.
