On the size of graphs without repeated cycle lengths

In 1975, P. Erdös proposed the problem of determining the maximum number f(n) of edges in a graph with n vertices in which any two cycles are of different lengths. In this paper, it is proved that f(n)≥n+1073t+73for t=1260r+169(r≥1) and n≥21194t2+87978t+159574. Consequently, lim infn→∞f(n)−nn≥2+765419071, which is better than the previous bounds 2 (Shi, 1988), 2.4 (Lai, 2003). The conjecture limn→∞f(n)−nn=2.4 is not true.