  The enumeration of independent sets of regular graphs is of interest in
statistical mechanics, as it corresponds to the solution of hard-particle
models. In 2004, it was conjectured by Fendleyet al. that for some rectangular
grids, with toric boundary conditions, the alternating number of independent
sets is extremely simple. More precisely, under a coprimality condition on the
sides of the rectangle, the number of independent sets of even and odd
cardinality always differ by 1. In physics terms, this means looking at the
hard-particle model on these grids at activity -1. This conjecture was recently
proved by Jonsson. Here we produce other families of grid graphs, with open or
cylindric boundary conditions, for which similar properties hold without any
size restriction: the number of independent sets of even and odd cardinality
always differ by 0, 1,-1, or, in the cylindric case, by some power of 2. We
show that these results reflect a stronger property of the independence
complexes of our graphs. We determine the homotopy type of these complexes
using Forman's discrete Morse theory. We find that these complexes are either
contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge
of spheres. Finally, we use our enumerative results to determine the spectra of
certain transfer matrices describing the hard-particle model on our graphs at
activity -1. These results parallel certain conjectures of Fendley et al.,
proved by Jonsson in the toric case.
