Combinatorial aspects of the non-symmetric strong spectral property for graphs

TL;DR

This paper explores the combinatorial conditions under which directed graphs enforce the non-symmetric Strong Spectral Property (nSSP), introducing new methods and resolving open questions about graph patterns and arc counts.

Contribution

It introduces a novel combinatorial approach to identify graph patterns requiring nSSP and addresses open problems about arc minimality in directed graphs.

Findings

Loop assignments in double paths are crucial for nSSP.

A negative answer is provided to an open question on irreducible tridiagonal patterns.

Confirmed the minimum number of arcs needed for nSSP in several graph families.

Abstract

In this paper, we investigate the non-symmetric Strong Spectral Property (nSSP) from a combinatorial perspective. To zero-nonzero patterns of matrices we associate directed graphs and study when they require or allow the nSSP, providing a framework that avoids verifying the nSSP for individual matrices. A new combinatorial method is introduced and used to recognise several patterns that require the nSSP. It is shown that loop assignments in double paths play a critical role in establishing this property, and we show that an open question regarding irreducible tridiagonal patterns has a negative answer. We also investigate whether the minimum number of arcs in a directed graph on $n$ vertices that requires the nSSP, is equal to $2n-1$, and confirm this minimum for several specific digraph families.