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arXiv · 2605.19707

On asymptotic values for the minimum number of spanning forests in simple regular graphs

Shaohan Xu·Kexiang Xu
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Abstract

Let $F(G)$ be the number of spanning forests in a graph $G$ and $\mathcal{C}(n,d)$ be the set of all connected $d$-regular simple graphs of order $n$. Define $\widehat{f}_{d}=\liminf_{n\rightarrow \infty}\{F(G)^{1/n}:G\in \mathcal{C}(n,d)\}$. Let $n_i$ be the number of vertices of degree $i$ in $G$. In this paper we give two lower bounds for $F(G)$ in terms of $n_i$ in connected graphs whose vertex degrees belong to $\{2,3\}$ and $\{2,3,4\}$, respectively. Furthermore, we determine the exact values of $\widehat{f}_3$ and $\widehat{f}_4$.

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Shaohan Xu, Kexiang Xu. 2026-05-19. On asymptotic values for the minimum number of spanning forests in simple regular graphs. https://arxiv.org/abs/2605.19707

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