Classifying 2-arc-transitive graphs of order a product of two primes

A classi cation of all arc transitive graphs of order a product of two primes is given Furthermore it is shown that cycles and complete graphs are the only arc transitive Cayley graph of abelian group of odd order Introductory remarks Throughout this paper graphs are nite simple and undirected By p and q we shall always denote prime numbers A k arc in a graph X is a sequence of k vertices v v vk of X not necessarily all distinct such that any two consecutive terms are adjacent and any three consecutive terms are distinct A graph X is said to be k arc transitive if the automorphism group of X denoted AutX acts transitively on the k arcs of X For a group G and a generating set S of G such that S S the Cayley graph Cay G S of G relative to S has vertex set G and edges of the form g gs g G s S For any vertex g of Cay G S and any subset S of S an S neighbour of g is a vertex of the form gs s S In the classi cation of arc transitive circulants that is Cayley graphs of cyclic groups was given Proposition A connected arc transitive circulant of order n n is one of the following graphs i the complete graph Kn which is exactly transitive ii the complete bipartite graph Kn n n which is exactly transitive iii the complete bipartite graph minus a matching Kn n n K n odd which is exactly transitive and iv the cycle Cn of length n which is k arc transitive for all k The arc transitive graphs from the above proposition will be referred to as trivial arc transitive graphs For other results on arc transitive graphs we refer the reader to Sec The object of this paper is to classify arc transitive graphs whose order is a product of two primes This will be done by a simple extension of some of the ideas in and by taking into account the known results on vertex transitive graphs of order a product of two primes The following is our main result Theorem A connected arc transitive graph of order pq where p and q are primes is isomorphic to one of the graphs in Table below where the two inside double lines divide the graphs into trivial ones and nontrivial ones with imprimitive or primitive automorphism group TABLE arc transitive graphs of order a product of two primes row qp valency the graph qp qp Kqp qp Cqp p p Kp p p p Kp p pK incidence graph of H nonincidence graph of H k n k kn k incidence graph of PG n k k n k k nonincidence graph of PG n k Petersen graph O O orb graph PSL orb graph PSL orb graph PSL Perkel orb graph PSL orb graph PSL orb graph M The proof of this theorem will be carried out over the next three sections In Section we analyze the case p q as a byproduct we prove that a arc transitive Cayley graph of an abelian group of odd order is either a cycle or a complete graph Proposition In Sections and the respective cases of AutX imprimitive and primitive are dealt with