The Diameter of Sparse Random Graphs

Abstract

We consider the diameter of a random graph G(n, p) for various ranges of p close to the phase transition point for connectivity. For a disconnected graph G, we use the convention that the diameter of G is the maximum diameter of its connected components. We show that almost surely the diameter of random graph G(n, p) is close to $\text{log}\text{n}\text{log(}\text{np}\text{)}$ if np → ∞. Moreover if $\text{np}\text{log}\text{n}\text{=}\text{c}\text{>}\text{8}$, then the diameter of G(n, p) is concentrated on two values. In general, if $\text{np}\text{log}\text{n}\text{=}\text{c}\text{>}\text{c}{}_0$, the diameter is concentrated on at most 2⌊1/c₀⌋ + 4 values. We also proved that the diameter of G(n, p) is almost surely equal to the diameter of its giant component if np > 3.6.