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, $\liminf\sb { n \to \infty} { f(n)-n \over \sqrt n} \geq \sqrt { 2 + { 2562 \over 6911}}.$
Categories: math.CO