A classification of regular embeddings of graphs of order a product of two primes

In [5], the classification problem of regular embeddings of a given graph was described in terms of pure group theory. With the philosophy in [5], we shall classify the regular embeddings of simple graphs of order pq for any two primes p and q (not necessarily distinct) in this paper (see Theorem 4.8). The classification is based on the direct analysis of the structure of the arc-regular subgroups with the cyclic stabilizers of the automorphism groups of such graphs. Our analysis is independent of the classification of primitive permutation groups of degree p or degree pq. It is also independent of the classification of the arc-transitive graphs of order pq (p 6 = q).