Dense and Symmetric Graph Formulation and Generation for Wireless Information Networks

Abstract

Dense and symmetric graphs are useful for modeling fast information distribution in wireless information networks. In this paper, we focus on a specific family of dense and symmetric graphs, the Borel Cayley graphs. More specifically, we investigate the various parameters in the original formulation of Borel Cayley graphs defined in the matrix domain. By eliminating redundant parameters, we propose a new and simpler formulation of Borel Cayley graphs. This new formulation is defined in the integer domain and the group operation resembles the generation of pseudorandom numbers, hence the name pseudo-random formulation. Through the establishment of propositions and corollaries, we proved that certain parameters do not affect the diameter under a specific condition. This result provides a guideline in choosing appropriate generators and thus reducing the computation time in the search of "good" or "bad" generators. Using this new formulation, we also show that Borel Cayley graphs are isomorphic to a sub-class of Cayley graphs proposed by Dinneen. Finally, some guidelines for choosing generators to avoid disconnected graphs are also provided.