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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, $\liminf_{n\to\infty}\frac{f(n)-n}{\sqrt n}\geq\sqrt{2+\frac{2562}{6911}}$ .

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If$P$is a simplicial$d$-polytope then its$h$-vector ($h$_{0},$h$_{1}, …,$h$_{$d$}) satisfies $$ {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} $$ . Moreover, if$h$_{$r$−1}=$h$_{$r$}for some $$ r\leq \frac{1}{2}d $$ then$P$can be triangulated without introducing simplices of dimension ≤$d$−$r$.

Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices in which no two cycles have the same length. In 1975, Erdős raised the question of determining $f(n)$ [see J. A. Bondy and U. S. R. Murty, Graph theory with applications, Elsevier, New York, 1976; MR0411988 (54 #117)]. We prove that for $n\geq 36\cdot 5t^2-4\cdot 5t+1$ one has $f(n)\geq n+9t-1$ . We conjecture that $\liminf (f(n)-n)/\sqrt n\leq\sqrt 3$ .

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