Fibonacci and Lucas numbers arising from two-component spanning forests of wheel graphs

TL;DR

This paper establishes a bijection between spanning forests of wheel graphs and spanning trees of fan graphs, deriving explicit formulas involving Fibonacci and Lucas numbers through combinatorial and analytic methods.

Contribution

It introduces a new bijection and explicit formulas for two-component spanning forests in wheel graphs, linking combinatorics with Fibonacci and Lucas sequences.

Findings

Explicit formulas for spanning forests involving Fibonacci and Lucas numbers

A constructive bijection between wheel and fan graph forests

Unified combinatorial and analytic framework for graph spanning forests

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 $W_{n+1}$.