A 6-Regular Torus Graph Family with Applications to Cellular and Interconnection Networks

Abstract

This paper introduces a special graph family, together with an extensive characterization of some of its properties and two of its immediate applications. The graph denoted by T i,j, is a regular hexavalent toroidal graph. The topological features of T i,j include vertex symmetry, Hamiltonian decomposition, translation and rotation isomorphism. Topologically, the graph is a torus, while algebraically, it can also be

A 6-Regular Torus Graph Family with Applications to Cellular and Interconnection Networks

“…Note: One can obtain the same result by using Euclidean geometry [IrTh99]. However, the proof would be rather lengthy.…”

“…defines the diameter of the largest induced subgraph of T i,j which is planar [IrTh99]. This subgraph defines a planar hexagonal tile inscribed in the bigger hexagonal (toroidal) tile representing the T i,j graph.…”

“…Table 2 shows the results for a comparative study among different interconnection topologies, including the regular triangulated toroidal graph (RTT). The optimal mesh refers to a 4-regular triangulated toroidal graph with a maximal number of nodes for a given diameter and which has been introduced in [IrTh99]. As it can be inferred from Table 2, the RTT is more scalable than the hypercube and, moreover, for a network embedded on a surface of genus 1 (torus), the RTT (i.e., T i,i-1 ) has the maximum number of nodes possible for a given diameter and constant node degree 6.…”