Problem 72 Use inductive reasoning to make ... [FREE SOLUTION]

Chapter 14: Problem 72

Use inductive reasoning to make a conjecture that compares the sum of the degrees of the vertices of a graph and the number of edges in that graph.

Short Answer

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The conjecture is: The sum of the degrees of the vertices in a graph is always twice the number of edges. Notation: \( \sum degree(V) = 2|E| \). This is because, for each edge in the graph, two vertices gain one degree.

Step by step solution

Defining the problem

Before proceeding, identify and define the main components: vertices, edges, degree of vertices and look into an example graph. Note: A graph \( G \) is made up of vertices \( V \) and Edges \( E \). The degree of a vertex number of edges emanating from it.

Exploring a basic graph

Consider a simple graph \( G \) with two vertices connected by one edge, i.e., \( |V|=2 \) and \( |E|=1 \). The degree of each vertex will be 1. Therefore, the sum of the vertex degrees is \( |V| + |V| = 2|V| = 2 \). Compare this to the number of edges \( |E|=1 \).

Generalize a conjecture

Remember deductive reasoning requires making broad generalizations. From our observation in step 2, it's clear that the sum of the vertex degrees is twice the total number of edges, i.e., \( \sum degree(V) = 2|E| \). This is our induction-based conjecture.

Justify the conjecture

The justification of this conjecture is that every edge contributes exactly two to the total sum of the degrees of the vertices: every edge is attached to two vertices. This assertion holds true for all types of graphs either simple, multi-graphs or pseudo-graphs. Thus, the conjecture is valid and can be substantiated.

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91影视

Use inductive reasoning to make a conjecture that compares the sum of the degrees of the vertices of a graph and the number of edges in that graph.

Short Answer

Step by step solution

Defining the problem

Exploring a basic graph

Generalize a conjecture

Justify the conjecture

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