[Paper Review] Spanning Trees in Grid Graphs

Paul Raff|ArXiv.org|Sep 15, 2008

Advanced Graph Theory Research2 references3 citations

TL;DR

This paper presents a general method to derive linear recurrence relations for the number of spanning trees in grid graphs, cylinder graphs, and torus graphs formed by the product of a fixed graph G and a path or cycle. By applying the Cayley-Hamilton theorem to transition matrices derived from partition-based state representations, the authors compute exact recurrences, generating functions, and closed-form formulas, yielding 15 new integer sequences and improved recurrences for known cases, including $C_2(n)$, $C_3(n)$, and $K_3 \times P_n$. The approach is systematic and applicable to any graph $G$ via characteristic polynomials of large matrices indexed by set partitions.

ABSTRACT

Building on work by Desjarlais, Molina, Faase, and others, a general method is obtained for counting the number of spanning trees of graphs that are a product of an arbitrary graph and either a path or a cycle, of which grid graphs are a subclass. Results are obtained pertaining to recurrences obtained in this manner, and numerous new integer sequences are found.

Motivation & Objective

• To develop a general computational method for deriving linear recurrence relations for the number of spanning trees in graph products $G \times P_n$ and $G \times C_n$.
• To extend prior work by Desjarlais and Molina on $2\times n$ and $3\times n$ grid graphs to arbitrary graphs $G$.
• To provide minimal linear recurrences, generating functions, and closed-form expressions for spanning tree sequences.
• To compute new sequences and improve known recurrences for families such as $C_3(n)$ and $K_3 \times P_n$.
• To formalize the use of the Cayley-Hamilton theorem in conjunction with the transfer-matrix method for spanning tree enumeration.

Proposed method

• The method uses the Matrix Tree Theorem to compute the number of spanning trees as a determinant, but focuses on deriving recurrences rather than direct computation.
• It models the state of the rightmost column of a grid using set partitions of $[2k]$, representing connectivity components across the boundary.
• A transition matrix $A$ of size $B_{2k} \times B_{2k}$ is constructed, where $B_{2k}$ is the Bell number, encoding how spanning forests evolve across columns.
• The number of spanning trees in $G \times C_n$ is computed as $(A^n v) \cdot d_k$, where $v$ is an initial state vector and $d_k$ is a tree-counting vector for cylinder closure.
• The recurrence order is bounded by $B_{2k}$, and the characteristic polynomial of $A$ yields the minimal linear recurrence.
• The approach generalizes to any graph $G$ on $k$ vertices, enabling recurrence derivation for $G \times C_n$ and $G \times P_n$.

Experimental results

Research questions

• RQ1 What is the minimal linear recurrence governing the number of spanning trees in $k \times n$ grid graphs as $n \to \infty$?
• RQ2 How can the number of spanning trees in cylinder graphs $G \times C_n$ be systematically computed for arbitrary $G$?
• RQ3 What is the structure of the transition matrix that captures the evolution of connectivity states across columns in grid-like graphs?
• RQ4 Can the Cayley-Hamilton theorem be effectively applied to derive minimal recurrences for spanning tree sequences?
• RQ5 What new integer sequences emerge from this method, and how do they compare to known sequences?

Key findings

• The method successfully derives a 6th-order linear recurrence for $C_2(n)$: $T_n = 10T_{n-1} - 35T_{n-2} + 52T_{n-3} - 35T_{n-4} + 10T_{n-5} - T_{n-6}$, matching OEIS A006235.
• A new 18th-order recurrence is found for $C_3(n)$, with sequence $\{1, 70, 1728, 31500, 508805, \ldots\}$, not yet in OEIS.
• A new 10th-order recurrence is derived for $K_3 \times P_n$, with sequence $\{3, 318, 12960, 410700, 11870715, \ldots\}$, also not yet in OEIS.
• The generating function for $C_2(n)$ is $\frac{x(1+2x-10x^2+2x^3+x^4)}{(-1+5x-5x^2+x^3)^2}$, confirming the recurrence.
• The generating function for $K_3 \times P_n$ is $\frac{3x(1+48x-697x^2-2474x^3+9918x^4+62x^5-2045x^6+96x^7+5x^8)}{(-1+29x-145x^2+145x^3-29x^4+x^5)^2}$.
• The method proves that the number of spanning trees in $G \times C_n$ satisfies a linear recurrence of order at most $B_{2k}$, where $k = |V(G)|$.