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.

In this note we obtain a simple expression of any finote group by means of its generating set. Applying this result we partly solve a conjecture of Cayley graphs proposed by Babai and Seress. We also obtain some other conclusions on diameters on Cayley graphs.

Many real networks, including those in social, technological, and biological realms, are small-world networks. The two distin-guishing characteristics of small-world networks has been based on probabilistic methods, with a rather small number of researchers advocating deterministic models. In this paper, we further the study of deferministic small-world networks and show that Cayley graphs may be good models for such networks. Small-world networks based on Cayley graphs possess simple structures and signif-icant adaptability. The Cayley-graph model has pedagogical value and can also be used for desiging and analyzing communication and the other real networks.

Let n, s be positive integers such that 2 s<n and s 6= n 2. An undirected double-loop network G(n; 1, s) is an undirected graph (V,E), where V=Zn={0, 1, 2,..., n−1} and E={(i, i+1 (mod n)), (i, i+s (modn)) | i 2 Z}. It is a circulant graph with n nodes and degree 4. In this paper, the sufficient and necessary conditions for a class of undirected double-loop networks to be optimal are presented. By these conditions, 6 new optimal and 5 new suboptimal infinite families of undirected double-loop networks are given.

In this paper, a new routing algorithm is given for the shuffle-exchange permutation network (SEPn). The length of the path between any two nodes given by our algorithm is not more than 11/16n2+O(n), Le., the diameter of SEPn is most 11/16n2+O(n). This improves on a 1/8 (9n2-22n+24) routing algorithm described earlier by S. Latifi and P. K. Srimani. We also show that the diameter of SEPn is more than 2n2-n.