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Let $C(d,k)$ and $AC(d,k)$ be the largest order of a Cayley graph and aCayley graph based on an abelian group, respectively, of degree $d$ anddiameter $k$. When $k=2$, it is well-known that $C(d,2)\le d^2+1$ with equalityif and only if the graph is a Moore graph. In the abelian case, we have$AC(d,2)\le \frac{d^2}{2}+d+1$. The best currently lower bound on $AC(d,2)$ is$\frac{3}{8}d^2-1.45 d^{1.525}$ for all sufficiently large $d$. In this paper,we consider the construction of large graphs of diameter $2$ using generalizeddifference sets. We show that $AC(d,2)\ge \frac{25}{64}d^2-2.1 d^{1.525}$ forsufficiently large $d$ and $AC(d,2) \ge \frac{4}{9}d^2$ if $d=3q$, $q=2^m$ and$m$ is odd.

Cayley graphs of diameter two from difference sets