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, Schoutens and van Eerten, that for some regtangular grids, with cylindric or toric boundary conditions, the alternating number of independent sets is extremely simple. The conjectures in the toric case have been proved by Jakob Jonsson.

I will present some ongoing work joint with Mireille Bousquet-Mélou and Eran Nevo, were we study other families of grid graphs with similar properties, that is, the independence complex have reduced Euler characteristic, 0, 1, -1 or a power of 2. This turns out to reflect a stronger property. Using Forman's Discrete Morse theory one can prove that the independence complexes are either contractible, homotopy equivalent to a sphere, or in the cylindric case, to a wedge of spheres. From these enumerative results we may also deduce the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity -1.