# 1月7日 福州大学常安教授学术报告

常安，福州大学教授，博士生导师。1998年6月毕业于四川大学应用数学专业，获博士学位。目前兼任中国运筹会理事、中国数学会组合数学与图论专业委员会副主任（常务理事）。主要从事图论领域中图与超图的谱理论及应用研究。已在国内外专业期刊发表研究论文60多篇，参加了包括国家重点研究计划（973计划）项目课题、国家自然科学基金重点项目在内的十余项国家级科研项目的研究工作，并承担了多项包括国家自然科学基金面上项目和其他省级科研项目的研究工作。2004年获福建省科学技术二等奖。

An$r$-uniform hypergraph is linear if every two edges intersect in at most onevertex. Given a family of $r$-uniform hypergraph $\mathcal{F}$, the linearTur\'{a}n number ex$_r^{lin}(n,\mathcal{F})$ is the maximum number of edges of a linear$r$-uniform hypergraph on $n$ vertices that does not contain any member of $\mathcal{F}$as a subhypergraph. Let $F$ be a graph and $r\geq 3$ a positive integer. The $r$-expansion of $F$ is the $r$-graph$F^+$ obtained from $F$ by enlarging each edge of $F$ with $r-2$ new verticesdisjoint from $V(F)$ such that distinct edges of $F$ are enlarged by distinctvertices. In this talk, we first focus on the linear Tur\'{a}n problem of thebipartite hypergraph $K_{s,t}^+$, and present some bounds for the linearTur\'{a}n number of $K_{s,t}^+$ for $t\geq s\geq 2$. Then by establishing theconnection between spectral radius of the adjacency tensor and structuralproperties of a hypergraph, we prove that when $n$ is sufficiently large, thespectral radius $\rho (K_{r+1}^+ )$ of the adjacency tensor of $K_{r+1}^+$ isno more than $\frac{n}{r}$, i.e.,$\rho (K_{r+1}^+ )\leq \frac{n}{r},$ withequality if and only if $r|n$ and $H$ is a transversal design, where thetransversal design is the balanced $r$-partite $r$-uniform hypergraph such thateach pair of vertices from distinct parts are contained in one hyperedgeexactly. An immediate corollary of this result is that$ex_r^{lin}(n,K_{r+1}^+)= \frac{n^2}{r^2}$ for sufficiently large $n$ and$3|n$. This is the joint work with Guorong Gao, and Yuan Hou.