In 1975, P. Erd\H os 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, $mc(n)\leq \frac{15}{16}n.$ \par We make the following conjecture: \par \bigskip \noindent{\bf Conjecture.} $$\lim_{n \rightarrow \infty} {mc(n)\over n}= 0.$$

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In 1975, P. Erdős 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 $n\geq\frac{6911}{16}t^2+\frac{514441}{8}t-\frac{3309665}{16}$ . Consequently, $\liminf_{n\to\infty}\frac{f(n)-n}{\sqrt n}\geq\sqrt{2+\frac{2562}{6911}}$ .

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