讲座1：An improved bound for strongly regular graphs

Sims showed that there are for primitive strongly regular graphs with fixed smallest eigenvalue -m except for a finite number of them they belong two families of graphs. Later in 1979, Neumaier made this bound explicit. In this talk we will improve this bound by Neumaier. This is based on joint work with Chenhui Lv, Greg Markowsky and Jongyook Park.

讲座2：Directions of point sets in affine planes

An \emph{affine plane} of order $q$ is a $2-(q^2,q,1)$ design.The classical construction, denoted AG$(2,q)$, has a vector space $\mathbb F_q^2$ as point set, and the affine lines of the space as blocks (or \emph{lines}).The lines come in $q+1$ parallel classes, and to each parallel class we assign a \emph{slope} or \emph{direction}.A set $S$ of $q$ points is said to \emph{determine} a direction $d$ if some line with slope $d$ is spanned by two points of $S$.Note that $d$ is {\bf not} determined by $S$ if and only if every line with slope $d$ intersects $S$ in exactly one point.A classical result states that a set $S$ of $q$ points in AG$(2,q)$ that determines at most $\frac{q+1}2$ directions must be a translate of a vector subspace of $\mathbb F_q^2$ over some subfield of $\mathbb F_q$.

Recently, the problem was generalized to study sets of $k q$ points for some integer $k$.We say that $S$ is \emph{equidistributed} from direction $d$ if all lines with slope $d$ intersect $S$ in $k$ points, and we call $d$ a \emph{special} direction otherwise.Surprisingly, every translation plane (which includes all planes AG$(2,q)$) has a set with exactly 3 special directions.

In this talk, I will discuss some of these new result, and some other generalizations of the classical problem of determined directions.This is based on joint work with Bence Csajb\' ok, Tam\' as Sz\H onyi, and Zsuzsa Weiner.

讲座3：Steklov eigenvalues of graphs

In this talk, we will survey recent results on Steklov eigenvalues of graphs.

讲座4：On signed graphs with fixed smallest eigenvalue

Let $G$ be a graph with smallest eigenvalue $\lambda_{\min}(G)$. In $1973$, Hoffman showed that: $\rm{(i)}$ for any real number $\lambda\leq -1$, if $\lambda_{\min}(G)\geq \lambda$, then there exists a positive integer $t=t_{\lambda}$, such that $G$ is $\{K_{1,t},\widetilde{K_{2t}}\}$-free; $\rm{(ii)}$ for any integer $t$, if $G$ is $\{K_{1,t},\widetilde{K_{2t}}\}$-free, then there exists a positive integer $\lambda=\lambda_t$, such that $\lambda_{\min}(G)\geq \lambda$. In $2016$, Kim, Koolen and Yang  gave a structure theory for graphs with fixed smallest eigenvalue. \par

In this talk, I will present a generalization of these results to signed graphs. Let $(G,\sigma)$ be a signed graph with smallest eigenvalue $\lambda_{\min}((G,\sigma))$. We showed that: $\rm{(i)}$ for any real number $\lambda\leq -1$, if $\lambda_{\min}((G,\sigma))\geq \lambda$, then there exists a positive integer $t=t_{\lambda}$, such that $(G,\sigma)$ is $\{(K_{1,t},+), (K_t,-)$ $(\widetilde{K_{2t}},+),\widehat{K_{2t}}\}$-switching-free; $\rm{(ii)}$ for any integer $t$, if $(G,\sigma)$ is $\{(K_{1,t},+), (K_t,-),(\widetilde{K_{2t}},+)$, $\widehat{K_{2t}}\}$-switching-free, then there exists a positive integer $\lambda=\lambda_t$, such that $\lambda_{\min}((G,\sigma))\geq \lambda$. Moreover, we gave a structure theory for signed graphs with fixed smallest eigenvalue. In the end, I will introduce an application of our method on signed graphs with smallest eigenvalue greater than $-1-\sqrt{2}$. \par

This is based on a joint work with Prof. Koolen and Mr. Liu Jing-Yuan.

1. Jack Koolen (中国科学技术大学)

2. Sam Adriaensen (布鲁塞尔自由大学)

3. Huiqiu Lin (布鲁塞尔自由大学)

4. Meng-Yue Cao (中国科学技术大学)