The zero forcing polynomial 
of a finite simple
graph 
is the generating function in which the coefficient of 
counts the zero forcing sets
of size 
(Boyer et al. 2019). The least exponent with
a nonzero coefficient is the zero
forcing number.
Newton and Brimkov (2026) constructed an infinite family for which the coefficient sequence is not unimodal. Their 82-vertex example has
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The same family supplies counterexamples for the polynomials that count connected or total zero forcing sets. These require the set to induce a connected graph or a graph without isolated vertices, respectively.
