Techniques for determining equality of the maximum nullity and the zero forcing number of a graph

TL;DR

This paper investigates conditions under which the maximum nullity and zero forcing number of a graph are equal, introducing new parameters and methods to identify such graphs, exemplified by the Aztec Diamond graph.

Contribution

It introduces a new graph parameter as a lower bound for maximum nullity and studies various parameters to characterize when nullity equals zero forcing number.

Findings

Maximum nullity equals zero forcing number for Aztec Diamond graph.

New graph parameter acts as a lower bound for maximum nullity.

Connectivity and matrix-based methods help identify nullity-zero forcing number equality.

Abstract

It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph. In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters which bound the maximum nullity of a graph below. In particular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdi\'ere type parameter and the vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.