Find the number of connected components for each of the following graphs. (a) A graph with 8 vertices and 6 edges is shown. Vertex a is connected to vertex b and to vertex d. Vertex b is connected to vertex a and to vertex c. Vertex c is connected to vertex b and to vertex d. Vertex d is connected to vertex a and to vertex c. Vertex e is isolated. Vertex f is connected to vertex g. Vertex g is connected to vertex f and to vertex h. Vertex h is connected to vertex g.   (b) A graph with 6 vertices and 6 edges is shown. Vertex u is connected to vertex w and to vertex y. Vertex v is connected to vertex x and to vertex z. Vertex w is connected to vertex u and to vertex y. Vertex x is connected to vertex v and to vertex z. Vertex y is connected to vertex u and to vertex w. Vertex z is connected to vertex v and to vertex x.   (c) A graph with 10 vertices and 8 edges is shown. Vertex a is connected to vertex b. Vertex b is connected to vertex a and to vertex d. Vertex c is connected to vertex c by a loop and to vertex i. Vertex d is connected to vertex b and to vertex e. Vertex e is connected to vertex d. Vertex f is connected to vertex g. Vertex g is connected to vertex f. Vertex h is connected to vertex i. Vertex i is connected to vertex c, to vertex h, and to vertex j. Vertex j is connected to vertex i.   (d) A graph with 4 vertices and 2 edges is shown. Vertex v1 is connected to vertex v3. Vertex v2 is connected to vertex v4. Vertex v3 is connected to vertex v1. Vertex v4 is connected to vertex v2.

For which of the following does there exist a simple graph G = - (V, E) satisfying the specified conditions? Select one: O A. It has 7 vertices, 10 edges, and more than two components. B. It has 8 vertices, 8 edges, and no cycles. O C. It has 6 vertices, 11 edges, and more than one component. O D. It is connected and has 10 edges, 5 vertices and fewer than 6 cycles. O E. It has 3 components, 20 vertices and 16 edges.

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

Question 4. Let K, be the complete graph on n vertices. How many edges does it have? Justify your answer.