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- 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.Help me please(1) A directed graph G is defined with the following adjacency matrix. Label the nodes of G as 1,2,3, 4 and plot G. (1 0 0 1 0 100 0 010 1001, (2) Let f(x) = x* + (-2)x² + 3x + 4. Re-write f(x) in the telescoping form that can be used in Horner's method.
- Consider the graph below with vertex set {a,b,c,d,e,f,g}. a Give the adjacency matrix of the graph with lexiographically ordered rows and columns.Consider the adjacency matrix 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 of a graph with nodes labeled a,b,c,d,e and f and lexiographically ordered rows and columns. 1 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 0 0. Suppose the above adjacency matrix identifies the network in a small company, where nodes represent workers and edges indicate that those workers interact during the week. The company has a single team leader and two different projects, one larger than the other. Workers in the same project interact often during the week. a) State which node represents the team leader, justifying your answer. b) State whether all workers are involved in at least one project and whether there is any worker, apart from the team leader, who works simultaneously on both projects. c) Explain what happens to the graph if the team leader suddenly resigns. Your response should be written in sentences.Consider the adjacency matrix 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 of a graph with nodes labeled a,b,c,d,e and f and lexiographically ordered rows and columns. 1 0 1 1 0 0 Lo 1 0 1 0 0 Suppose the above adjacency matrix identifies the network in a small company, where nodes represent workers and edges indicate that those workers interact during the week. The company has a single team leader and two different projects, one larger than the other. Workers in the same project interact often during the week. b) State whether all workers are involved in at least one project and whether there is any worker, apart from the team leader, who works simultaneously on both projects.
- III. Consider the directed graph described by the following:(a) Draw the graph.(b) Find a directed path from vertex 3 to vertex 6.(c) Find a directed cycle starting from and ending at vertex 4.(d) Find the adjacency matrix of the graph.(e) Does there exist a directed path from vertex 2 to vertex 6?5. Let G = (V, E) be a graph with vertex-set V = {1,2,3,4} and edge-set E = {(1, 2), (3, 2), (4, 3), (1, 4), (2,4)}. (a) Draw the graph. Find (b) maximal degree, i.e. A(G), (c) minimal degree, i.e. 8(G), (d) the size of biggest clique, i.e. w(G), (e) the size of biggest independent set, i.e. a(G), ter (f) the minimal number of colours needed to color the graph, i.e. x(G).1. 5 V1 V4 Represent the given graph with an V2 V5 V3 adjacency matrix.
- 1. (a) Find the adjacency matrix of the following graph: (c) Draw 5 5 5 5 5S V3. VA V3 V2 U1 (b) Find the incidence matrix of the graph with vertex set {1, 22, 23, 24} and edge set {1x2, 14, 223, 234, 24}. the graph whose incidence matrix is: V6 V5 1 1 1 0 00000 V2 10 0110000 0001 0 1 100 0000 1 1 010 0 1 0 0 0 V6 0 0 1 0 0 0 0 1 1 0 1 0 13) Let (V, E) be the graph with vertices a, b, c, d, e, f, and g, and edges ab, ac, bc, bd, be, cd, ce, de, af, df, ag, and eg. (a) Draw this graph. (b) Write down this graph's incidence table and its incidence matrix. (c) Write down this graph's adjacency table and its adjacency matrix. (d) Is this graph complete? Justify your answer. (e) Is this graph bipartite? Justify your answer. (f) Is this graph regular? Justify your answer. (g) Does this graph have any regular subgraph? Justify your answer. (h) Give an example of an isomorphism from the graph (V, E) to itself satisfying that p(a) = a. (i) Is the isomorphism from part (h) unique or can you find another isomorphism that is distinct from but also satisfies that (a) a? Justify your answer.Let G be a simple connected graph with n vertices and 1/2(n-1)(n-2)+2 edges. Use Ore's theorem to prove that G is Hamiltonian.