On asymptotic values for the minimum number of spanning forests in simple regular graphs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918516097024000 |
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| author | Xu, Shaohan Xu, Kexiang |
| author_facet | Xu, Shaohan Xu, Kexiang |
| contents | 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 $\widehat{f}_4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19707 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On asymptotic values for the minimum number of spanning forests in simple regular graphs Xu, Shaohan Xu, Kexiang Combinatorics 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 $\widehat{f}_4$. |
| title | On asymptotic values for the minimum number of spanning forests in simple regular graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.19707 |