发表机构
Nanjing University of Aeronautics and Astronautics; MIIT Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles; National Institute of Education, Nanyang Technological University（南京航空航天大学; 工业和信息化部航空飞行器数学建模与高性能计算重点实验室; 南洋理工大学国立教育学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究$C_4$-自由图的生成树数量极值，证明边数不超过$\frac12 q(q+1)^2$的$n$顶点图生成树数至多$n^{(n-3)/2}$，且对多数素数幂$q$，正交极性图达到该上界，证实London猜想。
AI 中文摘要
对于图$F$，Turán数$\ex(n,F)$是$n$个顶点上不含$F$的图的最大边数。设$q\ge 2$为整数，并令$n=q^{2}+q+1$。Brown、Erdős、Rényi和Sós独立证明了对于每个素数幂$q$，$\ex(n,C_{4})\ge \frac12 q(q+1)^{2}$，而Füredi随后在$q\notin\{1, 7,9,11,13\}$时建立了$\ex(n,C_{4})$的上界$\frac12 q(q+1)^{2}$。在本文中，我们证明每个具有至多$\frac12 q(q+1)^{2}$条边的$n$顶点$C_{4}$-自由图$G$满足$\tau(G)\le n^{(n-3)/2}$，其中$\tau(G)$表示$G$的生成树数量。特别地，对于每个素数幂$q\notin\{7,9,11,13\}$，上述$\tau(G)$的上界恰好由正交极性图达到，从而证明了London对所有此类$q$的猜想。
英文摘要
For a graph \(F\), the Turán number \(\ex(n,F)\) is the maximum number of edges in an \(F\)-free graph on \(n\) vertices. Let \(q\ge 2\) be an integer and set \(n=q^{2}+q+1\). Brown and Erdős, Rényi and Sós independently proved that $\ex(n,C_{4})\ge \frac12 q(q+1)^{2}$ for every prime power \(q\), and Füredi subsequently established the upper bound $\frac12 q(q+1)^{2}$ for $\ex(n,C_{4})$ whenever \(q\notin\{1, 7,9,11,13\}\). In this article, we prove that every \(C_{4}\)-free graph \(G\) on \(n\) vertices with at most \(\frac12 q(q+1)^{2}\) edges satisfies $τ(G)\le n^{(n-3)/2}$, where \(τ(G)\) denotes the number of spanning trees of \(G\). In particular, for every prime power $q\notin\{7,9,11,13\}$, the above upper bound on $τ(G)$ is attained precisely by the orthogonal polarity graphs, thereby proving London's conjecture for all such $q$.
