Some results on the structure and spectra of matrix-products

TL;DR

This paper investigates the structural and spectral properties of specific matrix-products linked to bipartite graphs, revealing characterizations of graph structures like caterpillars through spectral criteria, with implications for dynamical systems stability.

Contribution

It establishes new connections between matrix-product spectra and graph structures, including a characterization of caterpillars via spectral properties, advancing understanding in graph theory and dynamical systems.

Findings

Characterization of caterpillars through matrix-product spectra

Spectral properties linked to bipartite graph structures

Implications for stability in dynamical systems

Abstract

We consider certain matrix-products where successive matrices in the product belong alternately to a particular qualitative class or its transpose. The main theorems relate structural and spectral properties of these matrix-products to the structure of underlying bipartite graphs. One consequence is a characterisation of caterpillars: a graph is a caterpillar if and only if all matrix-products associated with it have real nonnegative spectrum. Several other equivalences of this kind are proved. The work is inspired by certain questions in dynamical systems where such products arise naturally as Jacobian matrices, and the results have implications for the existence and stability of equilibria in these systems.