Compatible Forts and Maximum Nullity of a Graph

TL;DR

This paper explores bounds on the maximum nullity of a graph using transversal numbers of compatible forts, extending existing theorems and addressing issues in matrix nullspace constructions.

Contribution

It generalizes theorems from symmetric to combinatorially symmetric matrices and introduces new methods for analyzing nullspaces via forts.

Findings

Generalized theorems from symmetric to combinatorially symmetric matrices

Developed special bases of nullspaces from transversal sets

Identified issues with minimal forts and how to avoid them

Abstract

We consider bounds on maximum nullity of a graph via transversal numbers of compatible collections of forts. Results include generalizations of theorems from symmetric to combinatorially symmetric matrices, special bases of matrix nullspaces derived from transversal sets, and examples of issues that arise when considering only minimal forts and how to avoid them. We also show an important difference between constructing symmetric and combinatorially symmetric matrices associated to a graph whose nullspaces are supported on collections of disjoint forts.