  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.
