For each of the following characteristics, draw a graph or explain why such a graph does not exist. a. Simple graph with seven nodes, each of degree 3 b. Four nodes, two of degree 2 and two of degree 3 c. Three nodes of degree 0, 1, and 3, respectively d. Complete graph with 4 nodes each of degree 2

Use the graph below to answer the follow questions. A. Determine how many vertices the graph has.      B. Determine how many edges the graph has.         C. Determine the degree of each vertex. vertex A:        vertex B:        vertex C:        vertex D:        vertex E:        vertex F:        vertex G:        vertex H:       D.This graph is currently connected. What edges could be removed to make it disconnected?

Q17. Consider the graph with the following vertices, edges, and weights (weights are stated as the third entry for each edge in E, with the first two entries representing the vertices joined by the edge): V = {a, b, c, d, e, f, g} E = {{a, b, 10}, {a, c, 12}, {a, d, 15}, {a, f, 3}, {b, c, 11}, {b, e, 9}, {b, f, 7}, {c, d, 11}, {c, g, 15}, {d, e, 12}, {d, f, 10}, {f, g, 16}} b a Your Answer: d Answer CD e What is the total weight of a minimum spanning tree of the graph? g

Q17. Consider the graph with the following vertices, edges, and weights (weights are stated as the third entry for each edge in E, with the first two entries representing the vertices joined by the edge): V = {a, b, c, d, e, f, g} E = {{a, b, 10}, {a, c, 12}, {a, d, 15}, {a, f, 5}, {b, c, 11}, {b, e, 9}, {b, f, 2}, {c, d, 11}, {c, g, 12}, {d, e, 12}, {d, f, 10}, {f, g, 16}} b a d e CD What is the total weight of a minimum spanning tree of the graph?

In a high school class, there are students of different talents (ex. music, art, debate, sports, etc.). A committee needs to be formed consisting of the maximum number of students such that the committee is diverse – no two students with the same talent are in the committee. Transform this problem to a graph theoretic problem. What are the vertices of the graph? What are the edges of the graph? What do you have to find in the graph?

Consider the following two graphs G1 = {{a,b, c, d, e, f}, {ab, a f, ae, bc, bf, be, cd, cf, df, ef}}, and G2 = {{v1, v2, v3, v4, V5, V6}, {v1v2, v1V3, V1V4, V1 V5, V2V3, V2V6, V3V4, V3V5, V3 V6, V4V5}}, %3D (a) is the function b c d f a e U5 v2 V4 V3 a graph isomorphism? (b) can you construct a bijection of vertices that is not an isomorphism? explain. (c) provide a drawing that represents this isomorphism class.