This M. S.c. thesis is based on the following papers • Sinha D. and Sharma D., “absorption Cayley graph”, Electronic Notes in Discrete Mathemetics. ?? (????) ???-???. • Sinha D., Garg P. and Singh A., “ Some properties of unitary addition Cayley graphs”, NNTDM ??(?) (????) ??-??. Let G be a finite group and S be a subset of G such that S = S^(-?) and ? ?S. Then Cayley graph, denoted by Cay(G, S) with respect to S is a graph with vertex set G and edge set E(G, S) ={gh | hg^(-?) ?S}. It is easy to see that Cay(G, S) is |S|-regular. A special kind of Cayley graph is the unitary Cayley graph of additive group Zn, where n is a positive integer, with respect to S = U_n, the set of units of Z_n. Thus the vertex set of Cay(Z_n, U_n) is Z_n and there is an edge between two vertices x, y if and only if x - y is a unit in Z_n.