The Spectra Of Cayley Graph On Abel Group

Let X = X(G, S) be the Cayley graph of an Abel group G with respect to S,the inverse closed subset of G. A Cayley graph X = X(G, C) is called a circulantgraph if G = Z_n. The main results of this paper can be divided into three parts.The first part give the expression of spectrum of Cayley graph on Abel group,the second part determine a class of integral Cayley graph on Abel group, thethird part discuss the number of spanning trees of integral circulant graphs.In chapter one, we introduce the background, terminology and some impor-tant results.Chapter two is devoted to studying the spectrum of Cayley graph on Abelgroup. The spectrum of circulant graph is given by [1]. In [14], L. Lovasz de-termined the spectrum of graph with transitive automorphism group. In [2],L. Babai derived an expression for the spectrum of the Cayley graph X(G, S) interms of irreducible characters of the group G. However, their expressions in[14] and [2] for the spectrum of Cayley graph are not explicit expressions. LetG = Z_n1×Z_n2×…×Z_nt, we give a formula of the spectrum of the Cayley graphX(G, S) by an explicit expression of primitive roots of unity. In terms of theformula, we give a new explicit expression of the spectrum of the k-cubic Q_k.In chapter three, we study the integral graphs. In 1974, Harary and Schwenk[12] asked "Which graphs have integral spectra?" From the formula of spectrumof k-cubic Q_k we deduce that the k-cubic Q_k is also an integral graph. WasinSo in [16] completely determined the integral circulant graphs by which we givea sufficient condition of integral Cayley graph on Abel group G. We can't findother S∈G such that the Cayley graph on Abel group X(G, S) is integral, atlast we pose a problem, i. e., if the sufficient condition is also necessary condition?In chapter four, we study the number of spanning trees of integral circulantgraphs. The number of spanning trees in a graph is an important invariant.it is also an important measure reliability of a network. It is an interestingproblem to enumerate the spanning trees for some vertex-transitive graphs espe- cially for the circulant graphs. The number of spanning trees of circulant graphsX(Z_n, {±1,±2}), X(Z_n, {±1,±3}), X(Z_n, {±1,±4}), X(Z_n, {±1,±5}), X(Z_n,{±2,±3}), X(Z_n, {±2,±4}) and so on can be found in [10], [19], [20]. In this pa-per, we give the number of spanning trees of integral eirculant graph X(Z_n, S)and its line graph, where n=2p, p is a prime.