Zero forcing for inertia sets

TL;DR

This paper introduces a new variation of zero forcing to bound the inertia set of a graph, relating eigenvalue counts to combinatorial properties, thus advancing spectral graph theory.

Contribution

It proposes a novel zero forcing variation that bounds the maximum nullity for matrices with a specified number of negative eigenvalues, linking spectral properties to graph invariants.

Findings

Provides upper bounds for maximum nullity with q negative eigenvalues

Establishes limits on positive eigenvalues based on zero forcing variation

Enables lower bounds for the inertia set of a graph

Abstract

Zero forcing is a combinatorial game played on a graph with a goal of turning all of the vertices of the graph black while having to use as few "unforced" moves as possible. This leads to a parameter known as the zero forcing number which can be used to give an upper bound for the maximum nullity of a matrix associated with the graph. We introduce a new variation on the zero forcing game which can be used to give an upper bound for the maximum nullity of a matrix associated with a graph that has $q$ negative eigenvalues. This gives some limits to the number of positive eigenvalues that such a graph can have and so can be used to form lower bounds for the inertia set of a graph.