A graph-theoretic condition for irreducibility of a set of cone preserving matrices

TL;DR

This paper establishes a graph-theoretic criterion for determining the irreducibility of matrix sets that preserve a cone, linking algebraic properties to graph connectivity, with applications in biology and chemistry.

Contribution

It introduces a novel condition connecting bipartite digraph connectivity to K-irreducibility of K-quasipositive matrix sets, expanding understanding of matrix set properties.

Findings

K-irreducibility inferred from strong connectedness of bipartite digraphs

Matrix-sets defined via products exhibit this property

Applications demonstrated in biological and chemical systems

Abstract

Given a closed, convex and pointed cone K in R^n, we present a result which infers K-irreducibility of sets of K-quasipositive matrices from strong connectedness of certain bipartite digraphs. The matrix-sets are defined via products, and the main result is relevant to applications in biology and chemistry. Several examples are presented.