Algebraic graph theory is one of the important topics in mathematics that is interested for specialists in the field of algebra and graph theory in recent years. Algebraic graph theory is a combination of two strands. The first field is the study of algebraic objects associated with graphs. The second field is the use of tools from algebra to derive properties of graphs.One of the interesting algebraic graphs is the Cayley graph for a group and associate subset. A Cayley graph is a graph that encodes a group in a graph.In this book, we consider the graph namely Caym(G,S) by using column matrices which is a new generalization of usual Cay(G,S) and will state some results on it.The purpose of this book is to introduce the generalized Cayley graph and determine its structure in some special cases.In this book, we clarify some basic properties of the new graph and assigned the structure of Caym (G,S) when Cay(G,S) is the following graphs: complete graph Kn, complete bipartite graph Kn,n , complete 3-partite graph Kn,n,n , complete 4-partite graph Kn,n,n,n, cycle graph Cn, of length n and some other graphs for every m≥2.