### A Lower Bound for the Number of Edges in a Graph Containing No Two Cycles of the Same Length

#### Abstract

In 1975, P. Erdős proposed the problem of determining the maximum number $f(n)$ of edges in a simple 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 $n\geq {{6911}\over {16}}t^{2}+{{514441}\over {8}}t-{{3309665}\over {16}}$. Consequently, $\liminf _{n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + {2562 \over 6911}}.$