Fibonacci and Lucas numbers arising from two-component spanning forests of wheel graphs
Tsuyoshi Miezaki, Shunya Tamura
TL;DR
The paper studies conditioned two-component spanning forests on the wheel graph $W_{n+1}$ by establishing a constructive bijection with spanning trees of the fan graph $F_n$, and by applying effective-resistance formulas to obtain explicit counts. The bijection $\tau_{n+1} \leftrightarrow \mathscr{T}_n$ links forest components to fan-tree structures, yielding $|\tau_{n+1}|=\mathscr{T}(F_n)=f_{2n-2}$. Concurrently, using Barrett et al.'s all-minors matrix-tree theorem and Bapat–Gupta's resistance formulas, the authors derive $F_{W_{n+1}}(u\mid v)=f_{2k}(\ell_{2n}-2)-f_{2n}(\ell_{2k}-2)$ and specialize to $F_{W_{n+1}}(v_1\mid v_2)=2(f_{2n-1}-1)$, $F_{W_{n+1}}(v_1\mid v_3)=2(\ell_{2n-2}-3)$, and $F_{W_{n+1}}(v_1\mid v_c)=f_{2n}$. These results reveal a unified framework connecting combinatorial bijections with analytic resistance methods, and establish explicit Fibonacci/Lucas expressions for the counts. The work also relates the sequences to OEIS entries, illustrating the deep interplay between graph structure and integer sequences. Future work may extend the bijection and resistance-based counts to other graph families.
Abstract
In this paper, we present a constructive bijection between a conditioned spanning forest of the wheel graph $W_{n+1}$ and a spanning tree of the fan graph $F_n$. In addition, by applying the effective resistance formula obtained by Bapat and Gupta \cite{bapat-gupta}, we derive an explicit formula for the number of two-component spanning forests of $W_{n+1}$ in which two specified vertices $u$ and $v$ lie in distinct components. Based on this result, we obtain explicit formulas for the following three conditioned two-component spanning forests $F_{W_{n+1}}(v_1\mid v_2)$, $F_{W_{n+1}}(v_1\mid v_3)$, and $F_{W_{n+1}}(v_1\mid v_c)$. These formulas are $F_{W_{n+1}}(v_1\mid v_2)=2(f_{2n-1}-1)$, $F_{W_{n+1}}(v_1\mid v_3)=2(\ell_{2n-2}-3)$, $F_{W_{n+1}}(v_1\mid v_c)=f_{2n}$, where $f_i$ and $\ell_j$ denote the $i$-th Fibonacci number and $j$-th Lucas number, respectively. As these identities show, the enumerations naturally lead to formulas involving Fibonacci numbers and Lucas numbers. Taken together, these two approaches show a unified perspective. One is the constructive combinatorial bijection, and the other is the analytic method based on effective resistance. Together they provide a new integrated framework for studying the structure of spanning forests on $W_{n+1}$.