On generating functions and limit theorems connected with maximal independent sets in grid graphs

In this paper we study some quantitative characteristics of maximal independent sets in grid graphs using methods of combinatorial analysis, enumerative combinatorics, mathematical analysis and linear algebra. We obtain the explicit generating functions for the number of maximal independent sets in cylindrical and toroidal lattices of width 4, 5, 6. We prove that the limits of $mn$-th root of the number of (maximal) independent sets in rectangular, cylindrical and toroidal $m\times n$-lattices exist and that they are equal. Nobody studied the quantitative characteristics of maximal independent sets in grid graphs with respect to cylindrical and toroidal lattices before. Also nobody proved the existence of the limits of $mn$-th root of the number of maximal independent sets in grid graphs. Thus, our paper is a further development of enumerative combinatorics.