  Zero forcing is an iterative graph coloring process, where given a set of
initially colored vertices, a colored vertex with a single uncolored neighbor
causes that neighbor to become colored. A zero forcing set is a set of
initially colored vertices which causes the entire graph to eventually become
colored. In this paper, we study the counting problem associated with zero
forcing. We introduce the zero forcing polynomial of a graph $G$ of order $n$
as the polynomial $\mathcal{Z}(G;x)=\sum_{i=1}^n z(G;i) x^i$, where $z(G;i)$ is
the number of zero forcing sets of $G$ of size $i$. We characterize the
extremal coefficients of $\mathcal{Z}(G;x)$, derive closed form expressions for
the zero forcing polynomials of several families of graphs, and explore various
structural properties of $\mathcal{Z}(G;x)$, including multiplicativity,
unimodality, and uniqueness.
