A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive.By Folkman [J.Combin.Theory 3(1967),215-232],there is no semisymmetric graph of order 2p or 2p 2 for a prime p,and by Malni et al.[Discrete Math.274(2004),187-198],there exists a unique cubic semisymmetric graph of order 2p 3,the so called Gray graph of order 54.In this paper,it is shown that there is no connected cubic semisymmetric graph of order 4p 3 and that there exists a unique cubic semisymmetric graph of order 8p 3,which is a Z 2 × Z 2-covering of the Gray graph.
