On the number of edges in some graphs.

In 1975, P. Erdős 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 , ... , c n) is the cycle length distribution of a graph G with n vertices, where c i is the number of cycles of length i in G. Let f (a 1 , a 2 , ... , a n) denote the maximum possible number of edges in a graph which satisfies c i ≤ a i , where a i is a nonnegative integer. In 1991, Shi posed the problem of determining f (a 1 , a 2 , ... , a n) which extended the problem due to Erdős. It is clear that f (n) = f (1 , 1 , ... , 1). Let g (n , m) = f (a 1 , a 2 , ... , a n) , where a i = 1 if i ∕ m is an integer, and a i = 0 otherwise. It is clear that f (n) = g (n , 1). We prove that lim inf n → ∞ f (n) − n n ≥ 2 + 40 99 , which is better than the previous bounds 2 (Shi (1988)), and 2 + 7654 19071 (Lai (2017)). We show that lim inf n → ∞ g (n , m) − n n m > 2. 444 , for all even integers m. We make the following conjecture: lim inf n → ∞ f (n) − n n > 2. 444. [ABSTRACT FROM AUTHOR]