报告题目 (Title)：Minimum ranks of graphs (图的最小秩)
报告人 (Speaker)：Li Zhongshan 教授（佐治亚州立大学）
报告时间 (Time)：2025年11月12日（周三）11:00
报告地点 (Place)：腾讯会议：577-109-892
邀请人(Inviter)：谭福平
主办部门：理学院数学系
报告摘要(Abstract)：Let $G$ be a graph of order $n$ with vertex set $V= \{ 1, 2, \dots, n\}$ and edge set $E$. The set of all real symmetric matrices associated with $G$ (with unrestricted diagonal entries), denoted ${\cal S}(G),$ is defined by $$ {\cal S}(G) =\{ A=[a_{ij}]\in \mathbb R^{n\times n} \mid a_{ii} \text{ is arbitrary }, a_{ij} \neq 0 \text{ iff } \{ i,j\} \in E \} .$$ The minimum rank of $G$, denoted mr$(G)$, is given by $$\mbox{mr}(G) = \min \{ \text{rank}(A) \mid A\in {\cal S}(G)\} .$$ Let $G^c$ denote the complement of $G$. The graph complement conjecture asserts that $$ \text{mr}(G) +\text{mr}(G^c) \le n+2. $$ We survey some results on the minimum ranks of graphs and present a latest result on the graph complement conjecture.
