Applications of Cayley Graphs

This paper demonstrates the power of the Cayley graph approach to solve specific applications , such as rearrangement problems and the design of interconnection networks for parallel CPU's. Recent results of the authors for efficient use of Cayley graphs are used here in exploratory analysis to extend recent results of Babai et al. on a family of trivalent Cayley graphs associated with P SL 2 (p). This family and its subgroups are important as a model for intercon-nection networks of parallel CPU's. The methods have also been used to solve for the first time problems which were previously too large, such as the diameter of Rubik's 2 × 2 × 2 cube. New results on how to generalize the methods to rearrangement problems without a natural group structure are also presented. 1. Introduction Each finite group G, together with a generating set Φ, determines a directed graph called a Cayley graph. Once a Cayley graph has been constructed for G, it is possible to obtain algorithmic solutions to the following problems: describe a complete set of rewriting rules for G relative to some lexicographic plus length ordering on the words of Φ [9]; obtain a set of defining relations for G in terms of Φ [6]; and find a word in Φ of minimal length that represents a specified element of G. The last problem is called the minimal word problem for G. The solution of the minimal word problem provides an optimal strategy for many rearrangement problems, where the elements of the generating set have some physical significance. These include problems in communications, which can be viewed as token movements on graphs [11], as well as such popular puzzles as Rubik's cube.