A Study of the Generalized Petersen Graph and a Novel Graph Y

Abstract

This paper presents a detailed study of several properties of the generalized Petersen graph, which are considered important to the parallel architectures. Similar to the well-known Petersen graph, the generalized Petersen graph contains two cycles; all the nodes in one cycle are each linked to their counterparts in the other cycle. Each cycle can hold either exactly five nodes or any number of nodes greater than six, according to our study. Besides the detailed analyses of several important properties of the generalized Petersen graph, we also propose a shortest-path routing algorithm and a general method for its VLSI layout. The second major contribution of this paper is the proposition of a new graph, denoted as Y, developed in conjunction with a novel approach to graph expansion. This expansion method recursively utilizes the structure of a bipartite graph and substitutes each node with a cyclic supernode. When compared with the popular hypercube structure, the Y graph not only contains up to 25% more nodes, but also has smaller diameters. Furthermore, close comparisons with the generalized Petersen graph reveals that they are both regular, but the Y graph is more favorable for being less restrictive in properties like number of nodes, degree, and diameter. Its structural flexibility can easily be seen from the fact that the generalized Petersen graph can be derived from the Y graph with ease. We also develop for the Y graph a shortest-path routing algorithm.