On the enumeration of connected sets in finite cylindrical lattice graphs

TL;DR

This work advances the enumeration of connected sets, i.e., lattice animals, in finite Cartesian and cylindrical lattice graphs by developing a transfer-matrix framework that relates the count in $C_m\times P_n$ to the complete graph case $K_m\times P_n$, enabling explicit formulas for $N(C_m\times P_n)$ with $m\le 7$. It introduces a tight lower bound $N_L(G\times P_n)$ via a subclass of densely connected column-sets and proves an exact upper bound for the cylindrical case when $m\le 7$, with concrete results for $m=4,5$. The paper provides detailed recurrences for $N(C_4\times P_n)$ and $N(C_5\times P_n)$ and performs asymptotic analysis, yielding growth constants for several graphs (e.g., $c(P_3\times P_n)\approx 1.6694$, $c(C_4\times P_n)\approx 1.8014$, $c(C_5\times P_n)\approx 1.7877$) and suggesting limiting values around $1.7196$ and $1.7914$ for some cases. These results connect finite-region connectivity to asymptotic lattice-animal growth and advance exact counting methods in combinatorial graph theory.

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 $O(N(C_5\times P_n))=1.7877^{5n}$.

Figures (4)

• Figure 1: $C_4\times P_4$ and $C_5\times P_4$
• Figure 2: Examples for sets $C$ in $\mathcal{C}(C_5\times P_4)$, where $C$ is the set of solid vertices
• Figure 3: The graph $K_{4}\times P_{n}$
• Figure 4: An example of the case that Lemma \ref{['lem2']} can not deal with, where $C$ is the set of solid vertices

Theorems & Definitions (10)

• Theorem 1.2
• Theorem 2.1: Ma
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• Corollary 3.1
• proof
• Corollary 3.2
• Example 4.1
• Example 4.2