Independent Set Enumeration and Estimation of Related Constants of Grid Graphs and Their Variants

TL;DR

This paper uses tensor network algorithms to enumerate independent sets on grid graphs, analyzing surface and parity effects, and provides new bounds and estimates for related combinatorial constants, improving accuracy over previous results.

Contribution

It introduces tensor network contraction methods for enumeration and analysis of independent sets on various grid graphs, deriving bounds and estimating constants with higher precision.

Findings

Surface and parity effects influence enumeration and entropy.

New bounds for combinatorial entropy and surface effect coefficients.

More accurate estimates of the hard square entropy constant.

Abstract

We applied tensor network contraction algorithms to compute the hard-core lattice gas model, i.e., the enumeration of independent sets on grid graphs. We observed the influence of surface effect and parity effect on the enumeration (and entropy), and derived upper and lower bounds for both the combinatorics entropy and the coefficients of surface effect by numerical analysis. Additionally, we conducted corresponding calculations and analyses for triangular grid graphs, king graph, and cylindrical grid graph. We computed and analyzed their associated constants and compared how different adjacency and boundary conditions affect these constants. Our computational results have contributed substantial new terms to the OEIS sequence A089980, A027740, A219741, A226444, A245013 and A286513. In addition, we have provided fairly accurate estimates of the relevant constants through numerical analysis of the obtained results. Among them, our valuation of the hard square entropy constant is more accurate than existing results. And we conject that the surface effect of the periodic boundary of the cylindrical grid graph is $0$--its estimated value of coefficients is very close to $0$.