A LOWER BOUND OF THE NUMBER OF EDGES IN A GRAPH CONTAINING NO TWO CYCLES OF THE SAME LENGTH

Abstract: 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 $n geq frac{6911}{16}t^{2}+ frac{514441}{8}t- frac{3309665}{16}$.
Consequently, $ liminf sb {n to infty} {f(n)-n over sqrt n}
geq sqrt {2 + {2562 over 6911}}.$