Guo's index for some classes of matrices

TL;DR

This paper calculates Guo's index for certain classes of matrices, specifically permutative and circulant matrices, providing constructive methods and MATLAB examples to determine minimal spectral radii for realizability.

Contribution

It introduces a method to compute Guo's index for permutative and circulant matrices, expanding understanding of spectral radius bounds for these classes.

Findings

Guo's index for permutative matrices is derived.

Constructive methods for calculating Guo's index are provided.

MATLAB examples illustrate the application of the results.

Abstract

A permutative matrix is a square matrix such that every row is a permutation of the first row. A circulant matrix is a matrix where each row is a cyclic shift of the row above to the right. The Guo's index $\lambda_0$ of a realizable list is the minimum spectral radius such that the list (up to the initial spectral radius) together with $\lambda_0$ is realizable. The Guo's index of some permutative matrices is obtained. Our results are constructive. Some examples designed using MATLAB are given at the end of the paper.