On The Spectra Of Cayley Graphs Over Groups

The concept of Cayley graphs was introduced by Arthur Cayley in 1878 to explain the concept of abstract groups which are described by a set of generators.Because of its simple structure,high symmetry and various types,then it deepens the understanding of graphs and groups,making the relationship between them closer.Let G be a finite group,and S be a subset of G with 1(?)S,the Cayley digraph on G with respect to S,denoted by X(G,S),is the directed graph whose vertices are the elements of G and whose arc set is {(g,gs):g ∈G,s ∈ S}.If in addition,S is inverse-closed(i.e.S=S-1={s-1:s ∈ S}),then X(G,S)can be regarded as an undirected graph,called Cayley graph.Let D2n=<a,b|an=b2=1,bab=a-1)be the dihedral group of order 2n.At present,the spectrum of Cayley graphs on dihedral groups is mostly focused on Cayley(undirected)graphs.In this paper,we first study the spectrum of Hermitian adjacency matrix(H-spectrum)of Cayley digraphs X((D2n,S)on dihedral group D2n with |S|＝3,and then show that all Cayley digraphs X(D2p,S)with |S|=3 and p odd prime are Cay-DS,namely if,for any Cayley digraph X(D2p,T),X(D2p,T)and X(D2p,S)have the same H-spectrum implies that they are isomorphic.Semi-cayley graph is a generalization of Cayley graph.Relevant results of the spectrum and Laplacian spectrum of Semi-Cayley graphs have been studied in literature.In Chapter 3,a formula of the normalized Laplacian spectrum of semi-Cayley graphs over abelian groups is given.As applications of our main result,the formula of the normalized Laplacian spectrum are also given for two classes of semi-Cayley graphs.