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It is known that the number of vertices of a graph of diameter two cannotexceed $d^2+1$. In this contribution we give a new lower bound for orders ofCayley graphs of diameter two in the form $C(d,2)>0.684d^2$ valid for alldegrees $d\geq 360756$. The result is a significant improvement of currentlyknown results on the orders of Cayley graphs of diameter two.

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