Answered: Let G be a graph with n ≥ 2 vertices…

Let G be a graph with n ≥ 2 vertices x1, x2, . . . , xn, and let A be the adjacency matrixof G. Prove that if G is connected, then every entry in the matrix A^n−1 + A^nis positive.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 36EQ

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Question

Let G be a graph with n ≥ 2 vertices x1, x2, . . . , xn, and let A be the adjacency matrix
of G. Prove that if G is connected, then every entry in the matrix A^n−1 + A^n
is positive.

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