# Prove that a simple, connected graph with n nodes has at least n – 1 arcs. (Hint: Show that this…

Prove that a simple, connected graph with n nodes has at least n – 1 arcs. (Hint: Show that this can be restated as “A simple, connected graph with m arcs has at most m + I nodes.” Then use the second principle of induction on m.)

Prove that a simple graph with n nodes (n  2) and more than C(n – 1, 2) arcs is connected. (Hint: Use Exercises 62 and 66.)

Exercises 62 and 66

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.

Prove that a simple, connected graph with n nodes has at least n – 1 arcs. (Hint: Show that this can be restated as “A simple, connected graph with m arcs has at most m + I nodes.” Then use the second principle of induction on m.)

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