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.

The clustering cofficient C of a network, Which is a measure of direct conuectivity between neighbors of the various nodes, ranges from O (for no connectivity) to 1 (for full connectivity). We define ex-tended clustering coefficients C (h) of a small-world network based on nodes that are at distance h from a source node, thus generalizing distance-1 neighborhoods employyed in computing the ordinary cluster-ing coefficient C = C(1). Based on known results about the distance distribution Pb(h) in a network, that is, the probability that a randomly chosen pair of vertices have distance h, we derive and experimentally validate the law Ps(h)C(h)< clogN/N, where c is a small constant that seldom exceeds 1. This result is significant because it shows that the product Ps(h)C(h) is upper-bouded by a value that is considerably smaller than the product of maximum values for Pb(h) and C(h).

In this paper, we propose a new class of interconnection networks, called "biswapped networks" (BSNs). Each BSN is built of 2n copies of some n-node basis network using a simple rule for connectivity that ensures its regularity, modularity, fault tolerance, and algorithmic efficiency. In particular, if the basis network is a Cayley digraph then so is the resulting BSN. Our proposed networks provide a systematic construction strategy for large, scalable, modular, and robust parallel architectures, while maintaining many desirable attributes of the underlying basis network that comprises its clusters. We show how key parameters of a BSN are related to the corresponding parameters of its basis network and obtain a number of results on internode distances, Hamiltonian cycles, and node-disjoint paths. We also discuss the relationship between BSNs and swapped or OTIS networks.

The resistance distance ri j between two vertices vi and vj of a (connected, molecular) graph G is equal to the effective resistance between the respective two points of an electrical network, constructed so as to correspond to G, such that the resistance of any edge is unity. We show how ri j can be computed from the Laplacian matrix L of the graph G: Let L(i) and L(i, j) be obtained from L by deleting its i-th row and column, and by deleting its i-th and j-th rows and columns, respectively. Then ri j=detL(i, j)/detL(i).

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.