KPPY 59

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.

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 $G^\ast$.

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:

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

A Schur ring $\mathcal{A}$ over a $p$-group $H$ is called a $p$-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.

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.

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).