   Chapter 10.2, Problem 21ES ### 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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# Let A be the adjacency matrix for K3, the complete graph on three vertices. Use mathematical induction to prove that for each positive integer n, all the entries along the main diagonal of A” are equal to each other and all the entries that do not lie along the main diagonal are equal to each other.

To determine

To prove:

Use mathematical induction to prove that for each positive integer n, all the entries along the main diagonal of An are equal to each other and all the entries that do not lie along the main diagonal are equal to each other.

Explanation

Given information:

Let A be the adjacent matrix for K3, the complete graph on three vertices.

Proof:

The adjacency matrix A=[aij] is n×n zero-one matrix with                        aij={ 1  if there s an edge from  v i  to  v j 0                                  otherwiseA complete graph Kn(n1) is a simple graph with n vertices and anedge between every pair of vertices.

Given:

A Is the adjacency matrix for K3

K3 is a simple graph with 3 vertices and with an edge between every pair of vertices (but no loops).

To proof:

An=[abbbabbba] for some nonnegative integers a and b and for all integers n1

PROOF BY INDUCTION:

Let P(n) be "An=[abbbabbba] for some nonnegative integers a and b"

Basis step n = 1

K3 is a simple graph with 3 vertices and with an edge between every pair of vertices (but no loops).

Since K3 contains no loops, the adjacency matrix A of K3 will contain a 0 on the main diagonal elements.

Since there is an edge between every pair of distinct vertices of K3, the adjacency matrix A of K3 will contain a 1 on every non-diagonal element.

A=[ 0 1 1 1 0 1 1 1 0]Thus P(1) is true, since a=0 and b=1.

Inductive step:

Let P(k) be true, thus Ak=[ a b b b a b b b a] for some integers a and b.We need to prove that P(k+1) is true

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