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

TL;DR

This paper investigates the asymptotic behavior of the minimum number of spanning forests in simple regular graphs, providing bounds and exact values for specific degrees.

Contribution

It introduces new lower bounds for the number of spanning forests based on vertex degrees and determines exact asymptotic values for degree 3 and 4 graphs.

Findings

Derived two lower bounds for $F(G)$ based on vertex degree counts.

Calculated exact asymptotic values $\u00af}_3$ and $}_4$ for degree 3 and 4 graphs.

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 $\widehat{f}_4$.