  In 1975, P. Erd\H{o}s proposed the problem of determining the maximum number
$f(n)$ of edges in a graph on $n$ vertices in which any two cycles are of
different lengths. Let $f^{\ast}(n)$ be the maximum number of edges in a simple
graph on $n$ vertices in which any two cycles are of different lengths. Let
$M_n$ be the set of simple graphs on $n$ vertices in which any two cycles are
of different lengths and with the edges of $f^{\ast}(n)$. Let $mc(n)$ be the
maximum cycle length for all $G \in M_n$. In this paper, it is proved that for
$n$ sufficiently large, $mc(n)\leq \frac{15}{16}n$. We make the following
conjecture:
  $$\lim_{n \rightarrow \infty} {mc(n)\over n}= 0.$$
