Non-isomorphic subgraphs in random graphs

TL;DR

This paper analyzes the asymptotic behavior of the number of unlabelled induced subgraphs in random graphs, revealing how it varies with the probability parameter and establishing new bounds for regular graphs.

Contribution

It provides the first detailed asymptotic analysis of subgraph counts in Erdős–Rényi and random regular graphs across different probability regimes.

Findings

Number of subgraphs becomes exponential when p passes 1/n

Maximum exponential base reached when p ≫ 1/n

Subgraph count approaches 2^n when p passes 2ln n/n

Abstract

We establish the asymptotic behaviour of $\mu(G(n,p))$, the number of unlabelled induced subgraphs in the binomial random graph $G(n,p)$, for almost the entire range of the probability parameter $p=p(n)\in[0,1]$. In particular, we show that typically the number of subgraphs becomes exponential when $p$ passes $1/n$, reaches maximum possible base of exponent (asymptotically) when $p\gg 1/n$, and reaches the asymptotic value $2^n$ when $p$ passes $2\ln n/n$. For $p\gg \ln n/n$, we get the first order term and asymptotics of the second order term of $\mu(G(n,p))$. We also prove that random regular graphs $G_{n,d}$ typically have $\mu(G_{n,d})\geq 2^{c_d n}$ for all $d\geq 3$ and some positive constant $c_d$ such that $c_d\to 1$ as $d\to\infty$.