On the enumeration of connected sets in finite cylindrical lattice graphs

TL;DR

This paper develops enumeration formulas for connected sets in finite cylindrical lattice graphs and establishes bounds and asymptotic growth rates, linking combinatorics and structural connectivity in lattice systems.

Contribution

It extends enumeration methods to cylindrical lattice graphs up to size 7 and provides explicit bounds and asymptotic analysis for connected sets in Cartesian product graphs.

Findings

Derived enumeration formulas for $N(C_m\times P_n)$ with $m\le 7$

Established a tight lower bound for connected sets in $G\times P_n$

Performed asymptotic analysis showing exponential growth rates

Abstract

A connected set in a graph is a non-empty set of vertices that induces a connected subgraph. In an infinite lattice, a connected set is often referred to as a lattice animal, whose enumeration up to isomorphism is a classical problem in both combinatorics and statistical physics. In this paper, we focus on the enumeration of connected sets in finite lattice graphs, providing a link between combinatorial counting and structural connectivity in the system. For any positive integers $m,n$, let $N(P_m\times P_n)$ and $N(C_m\times P_n)$ denote the number of all connected sets in the $(m\times n)$-lattice graph $P_m\times P_n$ and $(m\times n)$-cylindrical lattice graph $C_m\times P_n $, respectively. In 2020, Vince derived enumeration formulas for $N(P_m\times P_2)$ and $N(C_m\times P_2)$, and highlighted the increasing difficulty of extending these calculation results to larger (cylindrical) lattice graphs. Recently, the authors of this paper have developed a method based on multi-step recurrence formulas to obtain the enumeration formula for $N(P_m\times P_n)$ with $m\le 4$. In this article, we apply a similar approach to derive the enumeration formula for $N(C_m\times P_n)$ with $m\le 7$. Further, for the general case, we establish an explicit and tight lower bound on the number of connected sets in the Cartesian product graph $G\times P_n $ for any connected graph $G$, by employing the transfer matrix method on a subclass of connected sets. Based on this, we perform an asymptotic analysis on several lattice graphs and show that $O(N(P_3\times P_n))=1.6694^{3n}$, $O(N(C_4\times P_n))=1.8014^{4n}$, and $O(N(C_5\times P_n))=1.7877^{5n}$.