Perspective topologies of multiprocessor computing systems based on the Cayley graphs of groups of period 4

The definition of the Cayley graph was given to the famous English mathematician Arthur Cayley in the XIX century to represent algebraic group defined by a fixed set of generating elements. At present, Cayley graphs are widely used both in mathematics and in applications. In particular, these graphs are used to represent computer networks, including the modeling of topologies of multiprocessor computer systems (MСS) - supercomputers. Since this direction is actively developed. This is due to the fact that Cayley graphs have many attractive properties such as regularity, vertex transitive, small diameter and degree at a sufficiently large number of vertices in the graph. For example, such a basic network topology as the “ring”, “hypercube” and “torus” are the Cayley graphs. One of the commonly used topologies MCS is a k-dimensional hypercube. This graph is given by k-generated Burnside groups of period 2, which is denoted B ( k,2 ). The group B ( k,2 ) has a simple structure and is equal to the direct product of k copies of the cyclic group of order 2. A generalization of the hypercube is a n-dimensional torus, which is generated by the direct product of n copies of the cyclic groups whose orders may be different. In this work, We researched the structure of the Cayley graphs of group B ( k,4 ) - Burnside k-generated groups of period 4 (and some their quotients). Then we compared these graphs with the corresponding toruses and hypercubes. The analysis is shown that the graphs B ( k,4 ) have better properties in comparison with hypercubes and toruses. Therefore, they deserve attention in the design of advanced topologies of MCS.