In this paper, we extend known relationships between Cayley digraphs and their subgraphs and coset graphs with respect to subgroups to obtain a number of general results on homomorphism between them. Intuitively, our results correspond to synthesizing altemative, more economical, interconnection networks by reducing the number of dimensions and/or link density of existing networks via mapping and pruning. We discuss applications of these results to well-known and useful interconnection netorks such as hexagonal and honeycomb meshes, including the derivation of provably correct shortest-path routing algorithems for such networks.

The resistance distance rij between two vertices vi and t,,j of a (connected, molecular) graph G is equal to the resistance between the respective two points of all electrical network, constructed so as to correspond to G, such that the resistance of ally two adjacent points is unity. We show how the matrix elements rij call be expressed in terms of the Laplacian eigenvalues and eigenvectors of G. In addition, we determine certain properties of the resistance matrix R=||rij||.

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.

Despite numerous interconnection schemes proposed for distributed multicomputing, systematic studies of classes of inter-processor networks, that offer speed-cost tradeoffs over a wide range, have been few and far in between. A notable exception is the study of Cayley graphs that model a wide array of symmetric networks of theoretical and practical interest. Properties established for all, or for certain subclasses of, Cayley graphs are extremely useful in view of their wide applicability. In this paper, we obtain a number of new relationships between Cayley (di) graphs and their subgraphs and coset graphs with respect to subgroups, focusing in particular on homomorphism between them and on relating their internode distances and diameters. We discuss applications of these results to well-known and useful interconnection networks such as hexgonal and honeycomb meshes as well as certain classes of pruned tori.

We consider the relationships between Cayley digraphs and their coset graphs with respect to subgroups and obtain some general results on homomorphism and broadcasting between them. We also derive a general factorization theorem on subgraphs of Cayley digraphs by their automorphism groups. We discuss the applications of these results to well-known interconnection networks such as the butterfly network, the de Bruijn network, the cube-connected cycles network and the shuffle-exchange network.