Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 14 - Graph Theory - 14.1 Graphs, Paths, and Circuits - Exercise Set 14.1: 72

Answer

By inductive reasoning, we can make a conjecture that the sum of the degrees of the vertices is double the number of edges in the graph.

Work Step by Step

I drew a graph with four vertices. Each vertex is connected to every other vertex with one edge. Since each vertex is connected to three other vertices with one edge each, the degree of each vertex is 3. The sum of the degrees of the vertices is 3+3+3+3 which is 12. The number of edges in the graph is 6, because there is one edge between each pair of vertices. By inductive reasoning, we can make a conjecture that the sum of the degrees of the vertices is double the number of edges in the graph.
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