题目： Spectral Stability Theorems for Hypergraphs and Applications
主讲人：康丽英 教授
时间：2025年12月15日（星期一）14:00
地点：#腾讯会议：596-371-793
主办单位：理学院
主讲人简介：
康丽英, 上海大学数学系教授。曾获得“上海市三八红旗手”，“上海市曙光学者” 称号，曾获得“上海大学吴兴华数学奖”。研究兴趣包括极值图论、图和超图的谱。 在《Journal of Combinatorial Theory, Series B》、《SIAM Discrete Mathematics 》、《Journal of Graph Theory》、《European Journal of Combinatorics》等重要学术期刊上发表学术论文180 余篇。主持国家自然科学基金项目多项，参加国家自然科学基金重点项目1 项，参加重大研究计划1 项。现担任中国运筹学会常务理事、中国工业与应用数学学会组合图论专业委员会秘书长、中国数学会组合图论分会理事。担任国际期刊《Discrete Mathematics, Algorithms and Applications》、 《Journal of the Operations Research Society of China》、《Communications on Applied Mathematics and Computation》和国内期刊《运筹学学报》编委。
摘要：
Spectral stability results are powerful tools for solving spectral extremal problems, which says roughly that a near-extremal (with respect to spectral radius) $n$-vertex $F$-free graph must be structurally close to the extremal graphs. Such stability results are crucial in resolving spectral Turán-type problems. In this talk, we present spectral stability results for hypergraphs and their applications. For $k \geq r \geq 2$, let $H_{k+1}^{(r)}$ denote the $r$-uniform hypergraph obtained from $K_{k+1}$ by enlarging each edge with a new set of $(r-2)$ vertices. Let $F_{k+1}^{(r)}$ be the $r$-uniform hypergraph with edges: $\{1,2,\ldots,r\}=[r]$ and $E_{ij} \cup \{i,j\}$ over all pairs $\{i,j\} \in \binom{[k+1]}{2} \setminus \binom{[r]}{2}$, where $E_{ij}$ are pairwise disjoint $(r-2)$-sets disjoint from $[k+1]$. We establish a general criterion that can obtain spectral stability results easily. Utilizing this criterion, we then derive spectral stability results for $H_{k+1}^{(r)}$ and $F_{k+1}^{(r)}$, respectively. Our results offer $p$-spectral analogues of the results by Mubayi-Pikhurko [J. Combin. Theory Ser. B, 97 (2007) 669-678] and Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220-225], and connect both hypergraph Turán theorem and hypergraph spectral Turán theorem in a unified form via the $p$-spectral radius. This is a joint work with Lele Liu, Zhenyu Ni, Jing Wang.
