In 1975, P. Erd\H{o}s 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 (c1β,c2β,β―,cnβ) is the cycle length
distribution of a graph G of order n where ciβ is the number of cycles of
length i in G. Let f(a1β,a2β,β―,anβ) denote the maximum possible
number of edges in a graph which satisfies ciββ€aiβ where aiβ is a
nonnegative integer. In 1991, Shi posed the problem of determining
f(a1β,a2β,β―,anβ) which extended the problem due to Erd\H{o}s, it is
clear that f(n)=f(1,1,β―,1). Let g(n,m)=f(a1β,a2β,β―,anβ),aiβ=1
for all i/m be integer, aiβ=0 for all i/m be not integer. It is clear
that f(n)=g(n,1). We prove that liminfnβββnβf(n)βnββ₯2+9940ββ, which is better than the previous bounds
2β (Shi, 1988), 2+190717654ββ (Lai, 2017). We show
that liminfnβββmnββg(n,m)βnβ>2.444β, for all even integers m. We make the following conjecture:
liminfnβββnβf(n)βnβ>2.444β.Comment: 9 page