Let G be the graph on vertices V1, V2, ..., , V8 with the following adjacency matrix: 1 0 0 0 0 0 1 1 0 1 0 0 0 01 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 0 000001 001 00010000 00000 00 0 00010 1 1 1 0000 Sketch all the graphs (up to isomorphism) on four vertices that are isomorphic to an induced subgraph of G (identifying a suitable induced subgraph in each case).
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- Show that the three points (x1,y1)(x2,y2) and (x3,y3) in the a plane are collinear if and only if the matrix [x1y11x2y21x3y31] has rank less than 3.Let G = (V, E) be a graph with vertex-set V = {1, 2, 3, 4, 5} and edge-setE = {(1, 2),(3, 2),(4, 3),(1, 4),(2, 4),(1, 3)}.(a) Draw the graph.Find (b) maximal degree, i.e. ∆(G),(c) minimal degree, i.e. δ(G),(d) the size of biggest clique, i.e. ω(G),(e) the size of biggest independent set, i.e. α(G), ter(f) the minimal number of colours needed to color the graph, i.e. χ(G).| Determine whether the two simple graphs with the following adjacency matrix are isomorphic or not? Check all criteria you know. (a) [0 0 1 0 0111 0], [0111001 00] (b) [0 110 0101 0011 0110 ], [0110 1101 1011 011 ]
- FB 5.2 Consider the lexicographically ordered adjacency matrix 0 1 1 1 0 0 1 0 10 0 1 10 1 1 1 A = 10 10 0 0 0 0 10 0 1 0 0 10 1 0 for a graph G with vertex set {a, b, c,d,e,f}. a) Sketch this graph and state the degrees of each of the nodes. b) State, with justification, whether this is a simple graph, or not.b.) Determine whether the graph is simple or not. If it is a simple the adjacency matrix of the graph and determine the no. of paths of length 2 that it has.Let G be a graph with vertex set V(G) = {v1, v2, V3, V4, V5, V6, V7} and edge set E(G) = {v1v2, V2V3, VZV4, V4V5, V4V1, V3V5, V6V1, V6V2, V6V4, V7V2, V7V3, V7V4} Let H be a graph with vertex set V (H)= {u1, U2, U3, U4, U5, U6, U7} and edge set E(H)={u1u2, U1U5, U2U3, U2U4, UQU5, U2U7, UZU6, UZU7, U4U5, UĄU6, U5U6, U6U7} Are the graphs G and H isomorphic? If they are, then give a bijection f : V (G) V(H) that certifies this, and if they are not, explain why they are not.
- A simple directed graph with vertices U1, U2, U3, U4, U5, U6, U7 has adjacency matrix 0 1 00 1 0 0 0 1 0 0 1 0 0 0 1 1 001 1 0 1. What is the order of the graph? order = 2. How many edges does the graph have? answer= 3.What is the in-degree of vertex V3? in-degree = 4. What is the out-degree of vertex V3? out-degree = 5. What is the in-degree of vertex v5? in-degree = 6. What is the out-degree of vertex v5? out-degree = A = 0 0 0 0 0 0 1 10 0 0 1 00 1 0 000 1 0 1 0 0Which of the following statements are true. Do not show your explanations. (d) Two graphs are isomorphic to each other if and only if they have the same adjacency matrix.(e) If T is a tree with e edges and n vertices, then e + 1 = n. (f) Petersen graph is not Hamiltonian graph.(b) Find the adjacency matrix for the following directed graph: O V₁ es e6 e4 ez V3 • V2 e3
- 3) 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 graph and e E E(G) Let H be the graph with V (H) = V (G) and E(H) = E(G)\ {e} Then e is a bridge of G if H has a greater number of connected components than G. Let G be the simple graph with V (G) = {u, v, w, x, y, z) and E(G) = {uy, vx, vz, wx, xz}. For each e E E(G), state whether e is a bridge of G. Justify your answer.Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a graph Gn whose vertex set is Vn. Include {gn, hn} as an edge of Gn if and only if gn and hn differ by two edges, i.e. you can obtain one from the other by moving a single edge. Tell us anything you can about the graph Gn. For example, (a) How many vertices does it have? (b) Is it regular (i.e. all vertices the same degree)? (c) Is it connected? (d) What is its diameter?