On vertex-uniprimitive non-Cayley graphs of order pq

TL;DR

This paper investigates the structure of non-Cayley vertex-transitive graphs of order pq, where p and q are odd primes, focusing on the primitiveness of automorphism groups and their socles.

Contribution

It establishes that the automorphism group acting primitively on such graphs is uniprimitive and provides conditions on p, q, and the socle T.

Findings

G is uniprimitive, primitive but not 2-transitive

Conditions on primes p and q derived

Information about the minimality of the socle T

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 $T=\soc(G).$