Maximal independent sets on a grid graph

TL;DR

This paper counts the maximal independent sets in grid graphs using a recursive matrix approach, providing exact formulas and asymptotic behavior, advancing combinatorial enumeration methods for lattice models.

Contribution

It introduces a recursive matrix-relation method to count maximal independent sets on grid graphs, extending the state matrix recursion algorithm to this problem.

Findings

Derived a recursive matrix relation for counting maximal independent sets

Provided the partition function with respect to the number of vertices

Analyzed the asymptotic behavior of the maximal hard square entropy constant

Abstract

An independent vertex set of a graph is a set of vertices of the graph in which no two vertices are adjacent, and a maximal independent set is one that is not a proper subset of any other independent set. In this paper we count the number of maximal independent sets of vertices on a complete rectangular grid graph. More precisely, we provide a recursive matrix-relation producing the partition function with respect to the number of vertices. The asymptotic behavior of the maximal hard square entropy constant is also provided. We adapt the state matrix recursion algorithm, recently invented by the author to answer various two-dimensional regular lattice model problems in enumerative combinatorics and statistical mechanics.