TETRAVALENT s-TRANSITIVE GRAPHS OF ORDER TWICE A PRIME POWER

A graph is s-transitive if its automorphism group acts transitively on s-arcs but not on (s + 1)-arcs in the graph. Let X be a connected tetravalent s-transitive graph of order twice a prime power. In this paper it is shown that s = 1, 2, 3 or 4. Furthermore, if s = 2, then X is a normal cover of one of the following graphs: the 4-cube, the complete graph of order 5, the complete bipartite graph K-5,K-5 minus a 1-factor, or K-7.7 minus a point-hyperplane incidence graph of the three-dimensional projective geometry PG(2, 2); if s = 3, then X is a normal cover of the complete bipartite graph of order 4; if s = 4, then X is a normal cover of the point-hyperplane incidence graph of the three-dimensional projective geometry PG(2, 3). As an application, we classify the tetravalent s-transitive graphs of order 2p(2) for prime p.