On edge-transitive distance-regular antipodal covers of complete graphs - Ludmila Tsiovkina

Title: On edge-transitive distance-regular antipodal covers of complete graphs

Abstract: A distance-regular antipodal cover of a complete graph is equivalently defined as a connected graph whose vertex set can be partitioned into a set of $n$ blocks (or fibres) of the same size $r\ge 2$ such that each block induces an $r$-coclique, the union of any two distinct blocks induces a perfect matching, and any two non-adjacent vertices lying in distinct blocks have exactly $\mu\ge 1$ common neighbours;  such a graph   will be briefly referred to as an $(n,r,\mu)$-cover.
Let  $\Gamma$ be  an edge-transitive   $(n,r,\mu)$-cover and $G=Aut(\Gamma)$. It can be easily seen that $G$ induces a $2$-homogeneous permutation group $G^{\Sigma}$ on the set $\Sigma$ of fibres of $\Gamma$,  which by combination of two well-known theorems of Kantor and Burnside is either almost simple or affine. In the talk, I will overview some recent results on classification of edge-transitive $(n,r,\mu)$-covers both in  almost simple and affine cases. In particular, I will show that non-bipartite edge-transitive   $(n,r,\mu)$-covers in almost simple case can be characterized (in essentially a uniform way) as basic graphs of certain schurian association schemes of quasi-simple groups.