Hexagonal mesh and torus, as well as honeycomb and certain other pruned torus networks, are known to belong to the class of Cayley groups which are node-symmetric and possess other interesting mathematical properties. In this paper, we use Cayley-graph formulations for the aforementioned networks, along with some of our previous results on subgraphs and coset graphs, to draw conclusions relating to internode distance and network diameter. We also use our results to refine, clarify, and unify a number of previously published properties for these networks and other networks derived from them.

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.

Many real networks, including those in social, technological, and biological realms, are small-world networks. The two distinguishing characteristics of small-world networks are high local clustering and small average internode distance. A great deal of previous research on 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 deterministic 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 significant adaptability. The Cayley-graph model has pedagogical value and can also be used for designing and analyzing communication and the other real networks.

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.

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.