On vertex-uniprimitive non-Cayley graphs of order pq

Abstract

Let $p$ and $q$ be distinct odd primes. Let $\Gamma=(V(\Gamma), E(\Gamma))$ be a non-Cayley vertex-transitive graph of order $pq.$ Let $G\leq \Aut(\Gamma)$ acts primitively on the vertex set $V(\Gamma)$. In this paper, we show that $G$ is uniprimitive which is primitive but not 2-transitive and we obtain some information about $p, q$ and the minimality of the Socle $T=\soc(G).$