The number of graphs of given diameter

TL;DR

This paper derives asymptotic formulas for counting labeled graphs of a given diameter, revealing structural differences between typical graphs in different diameter regimes.

Contribution

It provides new asymptotic formulas for the number of graphs with specified diameter and describes their typical structures for various diameter ranges.

Findings

Asymptotic formulas for graphs with diameter 2 < d < n - c_1(log n)

Typical graphs combine a path of length d and a highly connected block

Different structures emerge for graphs with diameter > n - c_2(log n)

Abstract

In this paper it is proved that there are constants 0< c_2< c_1 such that an asymptotic formula can be given for the the number of (labeled) n-vertex graphs of diameter d whenever n tends to infinity and 2 < d < n - c_1 (log n). A typical graph of diameter d consists of a combination of an induced path of length d and a highly connected block of size n-d+3. In the case d > n- c_2(log n) another asymptotic formula is calculated and the typical graph has a completely different snakelike structure.