Symmetric graphs of diameter 2 with complete normal quotients

Carmen Amarra (University of Western Australia, Australia)

A graph has \emph{diameter} 2 if it is not a complete graph and if every pair of nonadjacent vertices is joined by a path of length 2. Our general problem is to examine the overall structure of graphs which are both arc-transitive and of diameter 2 using normal quotients. We identify the basic graphs in the class of arc-transitive diameter 2 graphs to be those $\Gamma$ which satisfy one of the following (relative to some group $G$ of automorphisms of $\Gamma$): (i) $\Gamma$ does not have a nontrivial $G$-normal quotient, or (ii) all nontrivial $G$-normal quotients of $\Gamma$ are complete graphs. In this talk we discuss basic graphs which are of type (ii), in particular the case where there are at least 3 nontrivial normal quotients. For this case we can identify all the graphs that arise, using the classification of the transitive finite linear groups.