报告题目 (Title)：Edge connectivity, spanning tree packing number and eigenvalues of (multi-)graphs（图与多重图的边连通度、边不交生成树最大数目和特征值）
报告人 (Speaker)： 王力工 教授（西北工业大学）
报告时间 (Time)：2023年3月21日(周二) 14:30
报告地点 (Place)：腾讯会议室 8737775758
邀请人(Inviter)：王文环
主办部门：理学院数学系
报告摘要：A multigraph is a graph that may have multiple edges but does not contain loops. For a positive integer $t$, let $\mathcal{G}_{t}$ be the set of simple graphs (or multigraphs) such that for each $G\in\mathcal{G}_{t}$ there exist at least $t+1$ non-empty disjoint proper subsets $V_{1}, V_{2}, \ldots, V_{t+1}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1}\cup V_{2}\cup\cdots\cup V_{t+1})\neq\phi$ and edge connectivity $\kappa'(G)=e(V_{i},V(G)\setminus V_{i})$ for $i=1,2,\ldots,t+1$. Let $D(G)$ and $A(G)$ denote the degree diagonal matrix and adjacency matrix of a simple graph (or a multigraph) $G$, and let $\mu_{i}(G)$ be the $i$th largest eigenvalue of the Laplacian matrix $L(G)=D(G)+A(G)$. In this paper, we investigate the relationship between $\mu_{n-2}(G)$ and edge connectivity or spanning tree packing number of a (multi-)graph $G\in\mathcal{G}_{1}$, respectively. We also give the relationship between $\mu_{n-3}(G)$ and edge connectivity or spanning tree packing number of a (multi-)graph $G\in\mathcal{G}_{2}$, respectively. Moreover, we generalize all the results about $L(G)$ to a more general matrix $aD(G)+A(G)$ (or $aD(G)+bA(G)$), where $a,b$ are two real numbers such that $a \geq-1, b\neq 0$ and $\frac{a}{b}\geq-1$. This is a joint work with Yang Hu and Cunxiang Duan.
