On Vertex-Transitive Graphs of Order \(qp\)

The structure and the hamiltonicity of vertex-transitive graphs of order \(qp\), where \(q\) and \(p\) are distinct primes, are studied. It is proved that if \(q < p\) and \(\text{p} \not\equiv 1 \pmod{\text{q}}\) and \(G\) is a vertex-transitive graph of order \(qp\) such that \({Aut}G\) contains an imprimitive subgroup, then either \(G\) is a circulant or \(V(G)\) partitions into \(p\) subsets of cardinality \(q\) such that there exists a perfect matching between any two of them. Partial results are obtained for \(\text{p} \equiv 1 \pmod{\text{q}}\). Moreover, it is proved that every connected vertex-transitive graph of order \(3p\) is hamiltonian.