Tetravalent half-edge-transitive graphs and non-normal Cayley graphs
Let X be a vertex-transitive graph, that is, the automorphism group Aut(X) of X is transitive on the vertex set of X. The graph X is said to be symmetric if Aut(X) is transitive on the arc set of X. suppose that Aut(X) has two orbits of the same length on the arc set of X. Then X is said to be half-arc-transitive or half-edge-transitive if Aut(X) has one or two orbits on the edge set of X, respectively. Stabilizers of symmetric and half-arc-transitive graphs have been investigated by many authors. For example, see Tutte [Canad J Math 11 (1959), 621624] and Conder and Marui? [J Combin Theory Ser B 88 (2003), 6776]. It is trivial to construct connected tetravalent symmetric graphs with arbitrarily large stabilizers, and by Marui? [Discrete Math 299 (2005), 180193], connected tetravalent half-arc-transitive graphs can have arbitrarily large stabilizers. In this article, we show that connected tetravalent half-edge-transitive graphs can also have arbitrarily large stabilizers. A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in Aut(Cay(G, S)). There are only a few known examples of connected tetravalent non-normal Cayley graphs on non-abelian simple groups. In this article, we give a sufficient condition for non-normal Cayley graphs and by using the condition, infinitely many connected tetravalent non-normal Cayley graphs are constructed. As an application, all connected tetravalent non-normal Cayley graphs on the alternating group A6 are determined.  2011 Wiley Periodicals, Inc. J Graph Theory
half-edge-transitive graph cayley graph normal cayley graph