# KPPY 81

Seungjai Lee, Eun-Kyung Cho, Jongyook Park, Sejeong Bang, Mitsugu Hirasaka

Feb 21, 2017. 11:00am-6:00pm at PNU

Schedule | ||

11:00am - 11:50 |
Seungjai Lee NIMS | A survey on zeta functions of groups and rings |

12pm - 1:25 | Lunch | |

1:30 - 2:20 |
Eun-Kyung Cho PNU | n finite groups which induces only commutative proper Schur rings |

2:30 - 3:20 |
Jongyook Park KIAS | On 2-walk-regular graphs |

3:30 - 4:20 |
Sejeong Bang Yeungnam University | Determination of geometric distance-regular graphs |

4:30 - 5:20 |
Mitsugu Hirasaka PNU | On finite metric spaces with some properties on distances and triangles |

## Abstracts

Seungjai Lee

A survey on zeta functions of groups and rings

A survey on zeta functions of groups and rings

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.

Eun-Kyung Cho

n finite groups which induces only commutative proper Schur rings

n finite groups which induces only commutative proper Schur rings

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.

Jongyook Park

On 2-walk-regular graphs

On 2-walk-regular graphs

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.)

Sejeong Bang

Determination of geometric distance-regular graphs

Determination of geometric distance-regular graphs

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.

Mitsugu Hirasaka

On finite metric spaces with some properties on distances and triangles

On finite metric spaces with some properties on distances and triangles

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 $a_2=a_3=4$.