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