In 1975, P. Erd\"{o}s proposed the problem of determining the
maximum number $f(n)$ of edges in a graph of $n$ vertices in
which any two cycles are of different
lengths. In this paper, it is proved that $$f(n)\geq n+36t$$ for $t=1260r+169 \,\ (r\geq 1)$
and $n \geq 540t^{2}+\frac{175811}{2}t+\frac{7989}{2}$. Consequently,
$\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 +
{2 \over 5}},$ which is better than the previous bounds $\sqrt
2$ (see [2]), $\sqrt {2+{2562\over 6911}}$
(see [7]).