Constructions of cospectral graphs with different zero forcing numbers

TL;DR

This paper demonstrates that certain NP-hard zero forcing parameters are not determined by graph spectra, providing constructions of cospectral graphs with differing zero forcing numbers using various graph operations.

Contribution

It introduces new methods to construct cospectral graphs with different zero forcing numbers, including regular graphs and graphs with arbitrarily large differences in these parameters.

Findings

Cospectral graphs can have different zero forcing numbers.

Construction methods include graph products, joins, and switching.

Regular cospectral graphs can differ in zero forcing numbers significantly.

Abstract

Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper we show that several NP-hard zero forcing numbers are not characterized by the spectra of several types of associated matrices with a graph. In particular, we consider standard zero forcing, positive semidefinite zero forcing, and skew zero forcing, and provide constructions of infinite families of pairs of cospectral graphs which have different values for these numbers. We explore several methods for obtaining these cospectral graphs including using graph products, graph joins, and graph switching. Among these, we provide a construction involving regular adjacency cospectral graphs; the regularity of this construction also implies cospectrality with respect to several other matrices including the Laplacian, signless Laplacian, and normalized Laplacian. We also provide a construction where pairs of cospectral graphs can have an arbitrarily large difference between their zero forcing numbers.