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

This paper presents some results about the maximum number of vertices in primitive regular graphs with exponent 3. A computational experiment was conducted. We have found the numbers of primitive regular graphs with degree $p\le9$, number of vertices $n\le16$ and exponent 3 for each pair $(n,p)$. We have found the upper bound $n_p$ for the maximum number of vertices in primitive regular graphs with exponent 3 and degree $p$: $n_p\le3(p-1)+2(p-2)(p-1)+(p-2)^2(p+1)$. Also, we have found the exact value of the maximum number of vertices in primitive regular graphs with degree 3 and exponent 3, namely, $n_3=12$.