The Cayley graphs of Burnside groups of exponent $3$

Let $B_k=B(k,3)$ be the $k$-generator Burnside group of exponent $3$. Previously unknown Hall’s polynomials of $B_k$ for $k\leq 4$ are calculated. For $k>4$ polynomials are calculated similarly but their output takes considerably more space. Then using computer calculations for $2\leq k\leq 4$ were obtained diameters and average diameters of the Cayley graphs of $ B_k $ and their some factors generated by the symmetric generating sets. It is shown that these graphs have better characteristics than hypercubes. It can be concluded that the Cayley graphs of $ B_k $ deserve attention in the design of advanced topologies of multiprocessor computer systems.