发表机构
Institute for Basic Science; London School of Economics and Political Science（基础科学研究院; 伦敦政治经济学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了最小度至少为d的连通n顶点图的非同构生成树类数至少为K_{d,n-d}的对应值，并确定了唯一极值图，同时刻画了接近极值时的结构性质。
AI 中文摘要
对于图$G$，令$\tau_{\mathrm{iso}}(G)$表示其生成树的同构类数量。对于每个固定的$d\ge3$和所有足够大的$n$，我们证明每个连通的$n$顶点图$G$，若满足$\delta(G)\ge d$，则有\\[\tau_{\mathrm{iso}}(G)\ge \tau_{\mathrm{iso}}(K_{d,n-d})=A_dn^{d-1}+O_d(n^{d-2}),\\]其中$A_d>0$为显式常数，且$K_{d,n-d}$是唯一的最小化图。这证实了Bitonti、Michel和Scott的一个猜想，并将其推广到所有$d\ge3$。我们还证明了任何具有$O(n^{d-1})$种生成树类型的此类图，除有界个顶点外，其余顶点均具有相同的$d$个邻居。
英文摘要
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.
