Let G be a connected graph with n ≥ 2 vertices. Let A be the adjacency matrix of G. Prove that the diameter of G is the least number d such that all the non-diagonal entries of the matrix A are positive.

Let G be a connected graph with n ≥ 2 vertices. Let A be the adjacency matrix of G. Prove that the diameter of G is the least number d such that all the non-diagonal entries of the matrix A are positive.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 69EQ

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Question

Let G be a connected graph with n ≥ 2 vertices. Let A be the adjacency matrix of G.
Prove that the diameter of G is the least number d such that all the non-diagonal entries
of the matrix A are positive.

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