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.

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted reat attention. A related and longtime unsolved problem is: for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained: (1) for any k 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k, e, c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k 0, an infinite family of singular k-tight optimal DLN can be constructed.

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.

得到有限可解群及其某些子群的阶与其Fitting 子群的阶之间的若干关系。

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.