tailieunhanh - Báo cáo toán hoc:"Tetravalent non-normal Cayley graphs of order 4p"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Tetravalent non-normal Cayley graphs of order 4p. | Tetravalent non-normal Cayley graphs of order 4p Jin-Xin Zhou Department of Mathematics Beijing Jiaotong University Beijing 100044 . China j xzhou@bj Submitted Oct 21 2008 Accepted Sep 7 2009 Published Sep 18 2009 Mathematics Subject Classifications 05C25 20B25 Abstract 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 the full automorphism group of Cay G S . In this paper all connected tetravalent non-normal Cayley graphs of order 4p are constructed explicitly for each prime p. As a result there are fifteen sporadic and eleven infinite families of tetravalent non-normal Cayley graphs of order 4p. 1 Introduction For a finite simple undirected and connected graph X we use V X E X A X and Aut X to denote its vertex set edge set arc set and full automorphism group respectively. For u v G V X denote by u v the edge incident to u and v in X. A graph X is said to be vertex-transitive edge-transitive and arc-transitive or symmetric if Aut X acts transitively on V X E X and A X respectively. In particular if Aut X acts regularly on A X then X is said to be 1-regular. Let G be a permutation group on a set Q and a G Q. Denote by Ga the stabilizer of a in G that is the subgroup of G fixing the point a. We say that G is semiregular on Q if Ga 1 for every a G Q and regular if G is transitive and semiregular. Given a finite group G and an inverse closed subset S G G 1 the Cayley graph Cay G S on G with respect to S is defined to have vertex set G and edge set g sg g G G s G S . A Cayley graph Cay G S is connected if and only if S generates G. Given a g G G define the permutation R g on G by x xg x G G. Then R G R g g G G called the right regular representation of G is a regular permutation group isomorphic to G. It is well-known that R G Aut Cay G S . So Cay G S is vertex-transitive. In general a vertex-transitive graph X is isomorphic to a Cayley graph on a group G if and only if Supported by the .