(a) Determine the smallest positive value of n for which a simple graph on n vertices and 2n edges can exist. Give an example of such a graph for the smallest n. b) Let G be a simple graph with 20 vertices. Suppose that G has at most two com- ponents, and every pair of distinct vertices u and v satisfies the inequality that deg(u) + deg(v) > 19. Prove that G is connected.

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(a) Determine the smallest positive value of n for which a simple graph on n vertices
and 2n edges can exist. Give an example of such a graph for the smallest n.
(b) Let G be a simple graph with 20 vertices. Suppose that G has at most two com-
ponents, and every pair of distinct vertices u and v satisfies the inequality that
deg (u) + deg(v) > 19. Prove that G is connected.
Transcribed Image Text:(a) Determine the smallest positive value of n for which a simple graph on n vertices and 2n edges can exist. Give an example of such a graph for the smallest n. (b) Let G be a simple graph with 20 vertices. Suppose that G has at most two com- ponents, and every pair of distinct vertices u and v satisfies the inequality that deg (u) + deg(v) > 19. Prove that G is connected.
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