Affiliation: Shanghai University

**Title: On sesqui-regular graphs with fixed smallest eigenvalue **

**Abstract:** Let $\lambda\geq2$ be an integer. For strongly regular graphs with parameters $(v, k, a,c)$ and fixed smallest eigenvalue $-\lambda$, Neumaier gave two bounds on $c$ by using algebraic property of strongly regular graphs. Subsequently, we studied a new class of regular graphs called \emph{sesqui-regular graphs}, which contains strongly regular graphs as a subclass, and proved that for a given sesqui-regular graph with parameters $(v,k,c)$ and smallest eigenvalue $-\lambda$, if $k$ is very large, then either $c \leq \lambda^2(\lambda -1)$ or $v-k-1 \leq \frac{(\lambda-1)^2}{4} + 1$. This is joint work with Jack Koolen, Brhane Gebremichel and Jae Young Yang.

You can watch the talk here, and the slides are available here.

*This talk was given on September 12th, 2022*