On the independence complex of square grids

Abstract

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 Fendley et 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, 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

Certain Homology Cycles of the Independence Complex of Grids

“…Their method relies on the assumption that one of the generators of I is of the form (±m, ±m); as of today, it remains unclear whether it is possible to extend their method to arbitrary groups I . Bousquet-Mélou, Linusson, and Nevo [1] examined the homology of other square grids with free and cylindrical boundary conditions. Fendley and Schoutens [5] computed the (co-)homology of the independence complex of octagon-square and enneagon-triangle grids; see Sect.…”

Hard Squares for z = –1

“…This case is discussed by Bousquet-Mélou et al in [3], who calculate P N (x) for N = 1,... , 10. We have extended these calculations to N = 1,... , 12 and give the Table 2.…”

“…More recently, Jonsson ([9]) has investigated the effect of first rotating the lattice through 45 • before imposing toroidal boundary conditions, and Bousquet-Mélou et al ( [3]) have looked at the problem using free boundary conditions. Again they found that P N (x) is a product of simple factors.…”

Cores of Simplicial Complexes

“…The application motivating this method is in the study of the independence and dominance complexes of a graph, which have been extensively studied in the recent years [6,11,12,[18][19][20][21]27] because of their applications in graph theory, network analysis, and statistical mechanics. Our method allows us to obtain several results on the topology of the independence and dominance complexes.…”

“…In Sect. 6 we apply the method of the core to the dominance complex Δ of a forest F . We prove that Δ always collapses onto the boundary of a cross-polytope whose dimension equals the matching number of F .…”