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Báo cáo toán hoc:" Locally primitive normal Cayley graphs of metacyclic groups"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Locally primitive normal Cayley graphs of metacyclic groups. | Locally primitive normal Cayley graphs of metacyclic groups Jiangmin Pan Department of Mathematics School of Mathematics and Statistics Yunnan University Kunming 650031 P. R. China jmpan@ynu.edu.cn Submitted Mar 3 2008 Accepted Jul 28 2009 Published Aug 7 2009 Mathematics Subject Classifications 05C25. Abstract A complete characterization of locally primitive normal Cayley graphs of meta-cyclic groups is given. Namely let r Cay G S be such a graph where G Zm.Zn is a metacyclic group and m pl p22 pr such that p1 p2 pt. It is proved that G D2m is a dihedral group and val r p is a prime such that p p1 p1 - 1 p2 - 1 . pt 1 . Moreover three types of graphs are constructed which exactly form the class of locally primitive normal Cayley graphs of metacyclic groups. 1 Introduction Throughout the paper groups are finite and graphs are finite simple and undirected. For a graph r let V r denote its vertex set. For v G V r let r v denote the set of vertices which are adjacent to v. If r is regular then r v is called the valency of r and denoted by val r . A digraph r is called a Cayley digraph if there exist a group G and a subset S c G 1 such that its vertex set can be identified with G and two vertices u v are adjacent if and only if vu-1 G S. If further S S-1 s-11s G S then r is undirected and called Cayley graph. This Cayley digraph is denoted by Cay G S and the vertex of r corresponding to the identity element of group G is denoted by 1. Let r be a graph and let X be a group of automorphisms of r that is X 6 Autr. Then r is called X-vertex transitive or simply called vertex transitive if X is transitive on V r and r is called X-locally primitive or simply called locally primitive if Xv x G X vx v is primitive on r v for each vertex v. A 2-arc of r is a sequence u v w This work was partially supported by NNSF K1020261 and YNSF 2008CD060 . THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R96 1 of three distinct vertices such that v is adjacent to both u and w. Then r is .