Further mathematical properties of Cayley digraphs applied to hexagonal and honeycomb meshes

In this paper, we extend known relationships between Cayley (di)graphs 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 alternative, 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 networks such as hexagonal and honeycomb meshes, including the derivation of provably correct shortest-path routing algorithms for such networks. Index terms – Cayley digraphs, Cellular networks, Coset graphs, Diameter, Distributed systems, Homomorphism, Interconnection networks, Internode distance, Parallel processing, Routing. List of key notation Unless explicitly specified, all graphs in this paper are undirected graphs. • ≤ • Subgroup relationship •< • Normal subgroup relationship •/• Set of (right) cosets • × • Graph or set cross-product • The symbol “•” repeated i times (•,•) Edge 〈•〉 Cyclic group → Mapping ↔ Bijection ≅ Isomorphic to ≡ Congruent to Γ, Δ, Σ Graphs or digraphs φ Homomorphism 1 Identity element of a group Aut( ) Automorphism group Ck Cycle (ring network) of size k Cay( ) Cayley graph CCCq Cube-connected cycles of order q Cos( ) Coset graph dis( ) Distance function E( ) Edge set of a graph G, H Groups K, N Subgroups sign(e) −1 if e < 0; 0 if e = 0; +1 if e > 0 S, T Generator sets, subsets of G V( ) Vertex set of a graph Zq Cyclic group of order q Z Elem. abelian d-group of order d d q _______________________________ 1 Research supported by the Natural Science Foundation of China and Guangdong Province. 2 Department of Computer Science, South China University of Technology, Guangzhou 510641, China. E-mail: [email protected] 3 Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106-9560, USA. E-mail: [email protected]