Year: 2026
Type: preprint
Abstract: For a graph $G$, let $τ_{\mathrm{iso}}(G)$ denote the number of isomorphism classes of its spanning trees. For every fixed $d\ge3$ and all sufficiently large $n$, we prove that every connected $n$-vertex graph $G$ with $δ(G)\ge d$ satisfies \[τ_{\mathrm{iso}}(G)\ge τ_{\mathrm{iso}}(K_{d,n-d})=A_dn^{d-1}+O_d(n^{d-2}),\] for an explicit constant $A_d>0$, and $K_{d,n-d}$ is the unique minimizer. This confirms a conjecture of Bitonti, Michel and Scott and extends it to every $d\ge3$. We also show that any such graph with $O(n^{d-1})$ spanning-tree types has all but a bounded number of vertices with the same $d$ neighbours.
Source: arXiv (Cornell University)
Authors: Yan, Zhifei, Zhang, Lu-Ming
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Institute for Basic Science
Award: G2765210714
OpenAlex ID: W7214372595