In this short communicattion, we extend the known relation-ships between Cayley digraphs and their subgraphs and coset graphs with respect to subgroups and obtain some general results on homomor-phism and distance between them. Intuitively, Our results correspond to synthesizing alternative, more economical, interconnection networks by reducing the number of dimensions and/or link density of existing net-works via mapping and pruning. We discuss applications of these results to well-known and useful interconnection networks such as hexagonal and honey comb meshes.