报告题目：Spectral Sidorenko inequalities and edge-spectral supersaturation

报告人：李永涛

报告时间：2026年7月7日10:30-11:30

报告地点：科技园阳光楼 南815

邀请人：高国荣

邀请单位：福州大学数学与统计学院

报告内容简介：

We develop a spectral approach to Sidorenko-type inequalities and apply it to establish sharp edge-spectral supersaturation results. Let $H$ be a bipartite graph with $v$ vertices and $e$ edges, where $v\le e$, and write $M(G)=2e(G)$. We prove that Sidorenko's conjecture is equivalent to a spectral strengthening: $$ \hom(H,G)\ge M(G)^e |V(G)|^{v-2e} \quad \text{ if and only if }\quad \hom(H,G)\ge \lambda(G)^{2e-v}M(G)^{v-e}.$$ We also introduce an operator-norm certificate which, via the Riesz--Thorin interpolation, gives direct proofs of the spectral Sidorenko inequality in several cases. The converse direction in the equivalence theorem is proved by a tensor-power spectral regularization lemma. Our main result provides a unified framework to prove sharp asymptotic edge-spectral supersaturation results for degenerate bipartite graphs with the Sidorenko property, including complete bipartite graphs and even cycles. Let $S_{t-1,m}$ be the split graph with $m$ edges obtained by joining a clique $K_{t-1}$ to an independent set. For every $m$-edge graph $G$ with $\lambda(G)>\lambda(S_{t-1,m})$,$$\texttt{\#} K_{t,t}(G)\ge\Big(\frac{2^{-(t-1)^2}}{(t!)^2}-o(1)\Big)m^t \quad \text{and}\quad \texttt{\#}C_{2t}(G)\ge\Big(\frac{(t-1)!}{2t^t}-o(1)\Big)m^t.$$ Both constants are best possible: the first is attained asymptotically by random graphs, while the second is attained by split graphs. The supersaturation proofs combine spectral Sidorenko inequalities with a heavy-edge pruning process, a Perron-vector localized/delocalized dichotomy, and incidence-matrix inequalities. (This is a joint work with Wilson Lin, Hong Liu and Shengtong Zhang)

报告人简介：

李永涛，现为清华大学丘成桐数学科学中心博士后，合作导师为马杰教授。此前，他于2023年至2025年间在中南大学完成第一期博士后研究，合作导师为冯立华教授。2019年至2023年，他在湖南大学攻读博士学位，师从彭岳建教授。李永涛的研究方向为图谱理论与极值组合，在谱半径与子结构数目问题、给定边数的谱极值问题等前沿课题上，与合作者取得了一系列重要成果，在 《J. Combin. Theory Ser. B》 《J. Combin. Theory Ser. A> 《SIAM J. Discrete Math》 《European J.Combin.》《J. Graph Theory》《Adv. in Appl. Math.》《Electron. J. Combin.》等知名期刊发表图谱领城的研究论文20余篇。