About a criterion of equality to 3 for exponent of regular primitive graph

This paper presents some results related to finding criterion of equality to 3 for the exponent of regular primitive graph. Several necessary and several sufficient conditions are found. We show that no one of them could be criterion. For that purpose, we have also run a computation experiment for counting the number of primitive regular graphs with the exponent 3, that don't satisfied those conditions. As for a graph of the diameter 2, the following criterion is found for it: graph with diameter 2 is primitive with exponent 3 iff each vertex in that graph lies on at least one cycle of length 3 and there is at least one edge that does not lie on cycles with length 3.