题目：On spanning tree edge dependences of graphs

报告人：烟台大学 杨玉军教授

时间：2021年11月20号（周六）上午11:00--11:50

报告地点：腾讯会议ID：181 346 639

报告摘要：For a connected graph $G$, let $\tau(G)$ be the number of spanning trees of $G$, and for $e\in E(G)$, let $\tau_G(e)$ be the number of spanning trees of $G$ containing $e$. The quality $d_{G}(e)=\tau_{G}(e)/\tau(G)$ is called the spanning tree edge density of $e$, and $\mbox{dep}(G)=\max\limits_{e\in E(G)}d_{G}(e)$ is called the spanning tree edge dependence of $G$. Given a rational number $p/q\in (0,1)$, if there exists a graph $G$ and an edge $e\in E(G)$ such that $d_{G}(e)=p/q$, then we say the spanning tree edge density $p/q$ is constructible. More specially, if there exists a graph $G$ such that $\mbox{dep}(G)=p/q$, then we say the spanning tree edge dependence $p/q$ is constructible. In 2002, Ferrara, Gould, and Suffel asked which rational spanning tree edge densities are constructible and which rational spanning tree edge dependences are constructible. In 2016, by construction method, Kahl proved that for all $0

\frac{1}{2}$ but fails for $p/q\leq \frac{1}{6}$. More precisely, on one hand, we construct a family of planar graphs to show that all rational spanning tree edge dependences $p/q>\frac{1}{2}$ are constructible via planar graphs, but on the other hand, we show that the spanning tree edge dependence of a planar graph $G$ satisfies $\mbox{dep}(G)> \frac{1}{6}$, implying that all rational spanning tree edge dependences $p/q\leq \frac{1}{6}$ are not constructible.

报告人简介：杨玉军，烟台大学数学与信息科学学院教授、院长。烟台大学数学与信息科学学院教授、院长。山东省优秀青年基金获得者，山东省高等学校青创人才引育计划立项建设团队负责人，山东省高校黄大年式教师团队骨干成员。2009年博士毕业于兰州大学。主要研究领域为图论及其应用，在Combinatorica、European J. Combin.、Proc. Roy. Soc. A、Discrete Math.、Discrete Appl. Math.、Linear Algebra Appl.、J. Phys. A: Math. Theor.等国际权威杂志发表论文40余篇。先后主持国家自然科学基金4项，中国博士后科学基金特别资助和面上项目各1项。2015年获得山东省高校优秀科研成果奖一等奖。受美国Welch基金资助，两次赴美国德州农工大学盖文斯顿分校从事博士后研究。担任中国商业统计学会理事、中国工业与应用数学学会图论组合及其应用专业委员会委员、山东数学会理事、美国《数学评论》评论员。

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