In 1975, P. Erd\H os 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. The sequence $(c_1,c_2,\cdots,c_n)$ is the cycle length
distribution of a graph $G$ of order $n$， where $c_i$ is the
number of cycles of length $i$ in $G$. Let $f(a_1,a_2,\cdots,$
$a_n)$ denote the maximum possible number of edges in a graph which satisfies
$c_i\leq a_i$ where $a_i$ is a nonnegative integer. In 1991, Shi posed the
problem of determining $f(a_1,a_2,\cdots,a_n)，$ which extended
the problem due to Erd\H os, it is clear that $f(n)=f(1,1,\cdots,1)$.
Let $g(n,m)=f(a_1,a_2,\cdots,a_n),$ $a_i=1$ for all $i/m$ be integer, $a_i=0$ for all $i/m$ be not integer.
It is clear that $f(n)=g(n,1)$.
We prove that
$\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 +
\frac{40}{99}},$ which is better than the previous bounds $\sqrt
2$ (Shi, 1988), $\sqrt {2 +
\frac{7654}{19071}}$ (Lai, 2017).
We show that $\liminf_{n \rightarrow \infty} {g(n,m)-n\over \sqrt \frac{n}{m}}
> \sqrt {2.444},$ for all even integers $m$. We make the following conjecture:
$\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} > \sqrt {2.444}.$