KPPY 81

Over the last few decades, zeta functions of groups and rings have become important tools in various areas of asymptotic group and ring theory. In this talk, I will introduce their definitions with brief history of their developments and their applications. I shall also present recent discoveries, and discuss further problems to solve.

Consider a group ring $\mathbb{C}[G]$ and a partition $\mathcal{P} = \{D_0=\{e\}, D_1, \cdots, D_d\}$ of $G$. Let $D_i^{*} = \{g^{-1} \mid g \in D_i\}$ and $\bar{D_i}$ is the sum of all the elements of $D_i$ for all $i \in \{0, 1, \cdots, d\}$. Then a subalgebra of $\mathbb{C}[G]$ generated by $\bar{D_0}, \bar{D_1}, \cdots, \bar{D_d}$ is called a Schur ring over $G$ and is denoted by $\mathfrak{G} =( G ; \mathcal{P})$. In this talk, we classify a group $G$ such that every proper Schur ring over it is commutative.

A $t$-walk-regular graph is a common generalization of distance-regular graphs on the one hand and $t$-arc-transitive graphs on the other hand. We focus on the case where $t =2$ because $s$-walk-regular graphs are $t$-walk-regular if $s\geq t$ and many properties of distance-regular graphs can be generalized to $2$-walk-regular graphs. In this talk, we first introduce some known results that hold for $2$-walk-regular graphs but are not true in general for $1$-walk-regular graphs. And then we will generalize a known result of distance-regular graphs to the class of 2-walk-regular graphs. (This is joint work with Jack Koolen and Zhi Qiao.)

A non-complete distance-regular graph is called {\it geometric} if there exists a set $\mathcal{C}$ of Delsarte cliques such that each edge lies in exactly one clique in $\mathcal{C}$. In this talk we study how to determine given distance-regular graphs with large valency are geometric or not.

Let $(X,d)$ be a finite metric space with $|X|=n$. For a positive integer $k$ we define $A_k(X)$ to be the quotient set of all $k$-subsets of $X$ by isometry, and we denote $|A_k(X)|$ by $a_k$. The sequence $(a_1,a_2,\ldots,a_{n})$ is called the isometric sequence of $(X,d)$. In this talk we aim to characterize finite metric spaces with $a_2=a_3=4$.