# Exercises 58-64 refer to the complement of a graph. If G is a simple graph, the complement of G,…

Exercises 58-64 refer to the complement of a graph. If G is a simple graph, the complement of G, denoted G&#39;, is the simple graph with the same set of nodes as G, where nodes x-y are adjacent in G&#39; if and only if they are not adjacent in G.

58. Draw G&#39; for the graph of Figure 5.18a.

59. Draw K4. 60. Show that if two simple graphs G1 and G2 are isomorphic, so are their complements G. and G2.

61. A simple graph is self-complementary if it is isomorphic to its complement. Prove that in a self-complementary graph with n nodes (n > 1), n = 4k or n = 4k + 1 for some integer k. (Hint: Use the result of Exercise 19.)

*62. a. Prove that in any simple graph G with at least two nodes, if G is not connected, then G&#39; is connected. (Hint: If G is not connected, then G consists of a collection of “disjoint” connected subgraphs.)

b. Find a graph G where both G and G&#39; are connected, thus showing that the converse of part (a) is false.

63. Given an adjacency matrix A for a simple graph G, describe the adjacency matrix for G&#39;.

64. Prove that if IN| 11 in a simple, connected graph G, then not both G and G&#39; can be planar.

Exercise 19

Find an expression for the number of arcs in Kn and prove that your expression is correct.

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