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Graphs : Connectedness, Vertices and Edges

11. Let G be a graph with n>= 2 vertices.
a) Prove that if G has at least (n-1) + 1 edges the G is connected.
( 2 )

b) Show that the result in (a) is best possible; that is, for each n>= 2, prove there is a graph with (n- 1)
( 2 )
edges that is not connected.

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Solution Preview

Please see the attached file.

(a) Prove by contradiction:
We know the number of edges in a complete graph with n nodes is (because in a complete graph, every node is connected to other n-1 nodes, therefore the degree of every node is (n-1), and the total degree of ...

Solution Summary

Connectedness, Vertices and Edges are investigated. The solution is detiled and well presented.