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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.01319 |
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| _version_ | 1866918193083187200 |
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| author | Ma, Hongxia Jin, Xian'an Zhang, Meiqiao |
| author_facet | Ma, Hongxia Jin, Xian'an Zhang, Meiqiao |
| contents | 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 $O(N(C_5\times P_n))=1.7877^{5n}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_01319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the enumeration of connected sets in finite cylindrical lattice graphs Ma, Hongxia Jin, Xian'an Zhang, Meiqiao Combinatorics 05C30 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 $O(N(C_5\times P_n))=1.7877^{5n}$. |
| title | On the enumeration of connected sets in finite cylindrical lattice graphs |
| topic | Combinatorics 05C30 |
| url | https://arxiv.org/abs/2511.01319 |