The distribution of the maximum number of common neighbors in the random graph

TL;DR

This paper investigates the asymptotic distribution of the maximum number of common neighbors among k vertices in a random graph, showing it converges to a Gumbel distribution after normalization.

Contribution

It derives the limiting distribution for the maximum number of common neighbors in G(n,p), extending understanding of extremal properties in random graphs.

Findings

Normalized maximum common neighbors converges to Gumbel distribution

Explicit formulas for normalization sequences a_n and σ_n

Provides insights into extremal graph properties

Abstract

Let $\Delta_{k;n}$ be the maximum number of common neighbors of a set of $k$ vertices in $G(n,p)$. In this paper, we find $a_n$ and $\sigma_n$ such that $\frac{\Delta_{k;n}-a_n}{\sigma_n}$ converges in distribution to a random variable having the standard Gumbel distribution.