About the maximum number of vertices in primitive regular graphs with exponent equals $3$

Some results on the maximum number of vertices in primitive regular graphs with exponent $3$ are presented. We have found upper bound of this number depending on the degree $p: n_p \le p^3-p^2-3p+5$. Also, the exact value of the maximum number of vertices in primitive cubic graphs with exponent $3$ is given: $n_3 = 12$. A computation experiment has been conducted, and we have found the number of primitive regular graphs with degree $p \le 9$, number of vertices $n \le 16$ and exponent $3$ for each $(n,p)$ pair.