Automorphisms Of A Family Of Cayley Graphs

Let G be a finite group and S a subset of G such that S=S-1= {s-1|s∈S)and 1(?)S.The Cayley graph Cay(G,S)on G with respect to S is defined with vertex set G and edge set{(g,sg)|g∈G,s∈S}.For g∈G, R(g)is defined by:h(?)hg,(?)h∈G.The map R:9(?)R(g),(?)g∈G is an homomorphism,and its homomorphic image is the right regular representation of G,denote by R(G).Obviously, R(G)is a regular subgroup of Aut(Cay(G,S)).A Cayley graph Cay(G,S)is called normal if the right regular representation R(G) of G is a normal subgroup of Aut(Cay(G,s)).Let n be a positive integer.The graph Cn[2K1]is a lexicographic product of Cn and 2K1 with vertex set{xi,yi|i∈Zn}and edge set{{xi,xi+1},{yi,yi+1},{xi, yi+1},{yi,xi+1}|i∈Zn}.The work in this paper is mainly about the automorphism group of Cn[2K1]. We have the following conclusions:(1)Aut(Cn[2K1])is isomorphic to S42(?)Z2 for n=4,or to Z2n(?)D2n for n>4；(2)we have determined all pairs of group G and Cayley subset S of G satisfying Cn[2K1]≌Cay(G,S).