About primitive regular graphs with exponent 2

Primitive regular graphs with exponent 2 are considered. We refine the known result that the number of edges of an undirected $n$-vertex graph with exponent 2 must be at least $(3n-3)/2$ for odd $n$ and $(3n-2)/2$ for an even $n$. For regular $n$-vertex graph with exponent 2 and $n>4$, the minimal number of edges is $2n$.