# Maximum nullity and zero forcing of circulant graphs

## TL;DR

This paper investigates the zero forcing number of circulant graphs, providing bounds and characterizations that relate to the minimum rank and maximum nullity, with specific focus on bipartite, cubic, and torus product circulants.

## Contribution

It offers new bounds and characterizations for the zero forcing number of various circulant graphs, enhancing understanding of their minimum rank and nullity.

## Key findings

- Bounds on zero forcing number for bipartite circulants
- Characterizations of zero forcing number for cubic circulants
- Conditions for equality in zero forcing number and nullity

## Abstract

It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of certain circulant graphs, including some bipartite circulants, cubic circulants, and circulants which are torus products, to obtain bounds on the minimum rank and the maximum nullity. We also evaluate when the zero forcing number will give equality.