# KPPY 59

Ilia Ponomarenko, Reza Sharafdini, Kijung Kim, Kwang Ju Choi, Jongyook Park

Sept. 14, 2013. 11am-6:00pm at PNU

Schedule | ||

11:00am - 11:50 |
Ilia Ponomarenko Petersburg Department of V. A. Steklov, Institute of Mathematics | On abelian Schur groups |

12pm - 1:25 | Lunch | |

1:30 - 2:20 |
Reza Sharafdini Persian Gulf University | Szeged and PI indices of Thorn Graphs |

2:30 - 3:20 |
Kijung Kim Pusan National University | Commutative $p$-Schur rings over non-abelian groups of order $p^3$ |

3:30 - 4:20 |
Kwang Ju Choi NIMS | A Characterization of Almost All Minimial Not Nearly Planar Graphs |

4:30 - 5:20 |
Jongyook Park University of Science and Technology of China | Symmetric association schemes with light tails |

## Abstracts

Ilia Ponomarenko

On abelian Schur groups

On abelian Schur groups

A finite group $G$ is called Schur, if every Schur ring over $G$
is the transitivity module of an appropriate permutation group
that contains a regular subgroup isomorphic to $G$. In the first
talk I am going to present recent results on abelian Schur
groups that was obtained in a joint work with S.Evdokimov and
I.Kovacs. In the second talk I will discuss a new result obtained
in a joint work with A.Vasilév that any non-abelian Schur group
$G$ is metabelian.

Reza Sharafdini

Szeged and PI indices of Thorn Graphs

Szeged and PI indices of Thorn Graphs

A topological index of a graph $G$ is a number $Top(G)$ which is invariant
under graph isomorphism.
Let $e=uv$ be an arbitrary edge of a simple connected graph $G=(V,E)$. The number of vertices of $G$ that are closer to $u$ than $v$ is denoted by $n_u(e,G)$ ($m_v(e)$ is defined analogously). The distance between the edge $f=xy$ and the vertex
$u$ in $G$, denoted by $d_G(f,u)$, is define as
$d_G(f,u) = \min \Big\{ d_G(x,u),d_G(y,u)\Big\}$.
The number of edges of graph $G$ that are closer to $u$ than $v$ is denoted by $m_u(e,G)$.
The

**vertex-Szeged index**of a graph $G$ is denoted by $Sz(G)$ and defined as \begin{equation} Sz(G) = \sum_{e=uv} n_u(e,G)\,n_v(e,G) \ . \end{equation} The**edge-Szeged index**is obtained by replacing $n_u(e,G)\,n_v(e,G)$ in the above equation by $m_u(e,G)\,m_v(e,G)$. Hence the edge-Szeged index is given by $Sz_e(G) = \sum_{e=uv} m_u(e,G)\,m_v(e,G)$. The**edge-$PI$ index**and**vertex-$PI$ index**is defined respectively as $ PI_e = PI_e(G) = \sum_{e=uv} \Big[ m_u(e,G) + m_v(e,G) \Big]$ and $PI_v(G) = \sum_{e=uv} \Big[ m_u(e,G) + m_v(e,G) \Big]$. Let $G$ be a connected graph. The thorn graph $G^\ast=G^\ast\Big(p_u\mid u\in V(G)\Big)$ ($p_u\ge 0$ for all $u\in V(G)$) of $G$ is obtained from $G$ by attaching to each vertex like $u$, $p_u$ new vertices of degree one. The pendent vertices attached to the vertex $u$ are called thorns of $u$. Special cases of thorn graphs have been already considered by Cayley (1874) and later by Pólya (1937). Years later, Gutman (2010) established relations between the Wiener indices of $G^\ast$ and $G$ for a connected graph $G$. Since then several study of different topological indices of general and some particular thorn graphs and trees like Wiener number (Zhou, Vukičević 2009), modified Wiener index (Vukičević, Graovac, 2004), variable Wiener index (B. Zhou et.al. 2006), altered Wiener index (Vukičević et. al. 2007), terminal Wiener index (Heydari, Gutman 2010), Hosoya polynomial (Walikar et.al. 2006), Schultz index (Vukičević et. al. 2005), have already been considered. In this talk we aim to compute these mentioned indices for thorn graph $G^\ast$. Kijung Kim

Commutative $p$-Schur rings over non-abelian groups of order $p^3$

Commutative $p$-Schur rings over non-abelian groups of order $p^3$

Let $H$ be a finite group.
We denote by $\mathbb{C}H$
the group algebra of $H$ over the complex number field $\mathbb{C}$.
For nonempty subset $T \subseteq H$,
we set $\underline{T}:= \sum_{t \in T}t$.
A subalgebra $\mathcal{A}$ of
the group algebra $\mathbb{C}H$ is called a

*Schur ring*over $H$ if the following conditions are satisfied:- there exists a basis of $\mathcal{A}$ consisting of $\underline{T_0} = 1_H, \underline{T_1}, \dotsc, \underline{T_r}$;
- $\bigcup_{i=0}^r T_i = H$ and $T_i \cap T_j = \emptyset$ if $i \neq j$;
- for each i, there exists $i'$ such that $T_{i'} = T_i^{-1}:=\{ t^{-1} \mid t \in T_i \}$.

*$p$-Schur ring*if the size of every element in $\{ T_0, T_1, \dotsc, T_r \}$ is a power of $p$. In this talk, we study the Schurity problem of commutative $p$-Schur rings over non-abelian groups of order $p^3$. In particular, we characterize a class of non-Schurian $p$-Schur rings. Kwang Ju Choi

A Characterization of Almost All Minimial Not Nearly Planar Graphs

A Characterization of Almost All Minimial Not Nearly Planar Graphs

In this talk, we define {\it nearly planar} graphs, that is, graphs
that are edgeless or have an edge whose deletion results in a planar
graph. We show that all but finitely many graphs that are not nearly
planar and do not contain one particular graph have a well-understood
structure based on large M\"{o}bius ladders.

Jongyook Park

Symmetric association schemes with light tails

Symmetric association schemes with light tails

In this talk we generalize light tail properties for distance-regular graphs to certain symmetric association schemes (Joint work with M. Camara, E. R. van Dam and J.H. Koolen).