For any $d\ge 11$ we construct graphs of degree $d$, diameter $2$, and order $\frac{8}{25}d^2+O(d)$, obtained as lifts of dipoles with voltages in cyclic groups. For Cayley Abelian graphs of diameter two a slightly better result of $\frac{9}{25}d^2 + O(d)$ has been known \cite{MSS} but it applies only to special values of degrees $d$ depending on prime powers.
the degree-diameter problem, voltage assignment and lift, dipole
05C12, 05C35