  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 $$f(n)\geq n+32t-1$$ for
$t=27720r+169 (r\geq 1)$ and $n\geq{6911/16}t^{2}+{514441/8}t-{3309665/16}$.
Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 +
{2562 \over 6911}}.$
