   Chapter 10.1, Problem 55ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# What is the maximum number of edges a simple disconnected graph with n vertices can have? Prove your answer.

To determine

What is the maximum number of edges a simple disconnected graph with n vertices can have? Prove your answer.

Explanation

Given information:

A simple disconnected graph with n vertices.

Calculation:

Let G be a simple disconnected graph with n vertices.

In an exercise of the previous section, we proved that the maximum number of edges of a simple graph with n edges is n(n1)2 (as the complete graph Kn contains n(n1)/2 edges).

However, in this case, we know that the graph G is also disconnected, which implies that there need to exist two vertices between which there exists no walk. Let us consider one vertex v in G. We then need to remove at least n − 1 edges from the complete graph Kn to disconnect vertex v from the rest of the graph (as the vertex v is connected to n − 1 other vertices by an edge)

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