On applications of the Cayley graphs of some finite groups of exponent five

Let $B_0(2,5)$ be the largest two–generator finite Burnside group of exponent five. It has the order $5^{34}$. We define an automorphism $\varphi$ which translates generating elements into their inverses. Let $C_{B_0(2,5)}(\varphi)$ be the centralizer of $\varphi$ in $B_0(2,5)$. It is known that $|C_{B_0(2,5)}(\varphi)|=5^{16}$. The growth functions of the centralizer are computed for some generating sets in the article. As the result we got diameters and average diameters of corresponding the Cayley graphs of $C_{B_0(2,5(\varphi)}$.