Cayley graphs of diameter two with order greater than 0.684 of the Moore bound for any degree

It is known that the number of vertices of a graph of diameter two cannot exceed d2+1d2+1. In this contribution we give a new lower bound for orders of Cayley graphs of diameter two in the form C(d,2)>0.684d2C(d,2)>0.684d2 valid for all degrees d≥360756. The result is a significant improvement of currently known results on the orders of Cayley graphs of diameter two.