Asymptotic study of subcritical graph classes

TL;DR

This paper introduces a unified method for asymptotic enumeration and property analysis of subcritical graph classes, revealing universal growth patterns and normal degree distributions in large random graphs.

Contribution

It provides a general approach applicable to various subcritical graph classes, unifying their asymptotic enumeration and probabilistic properties analysis.

Findings

Number of graphs grows as n^{-5/2} * gamma^n with computable constants.

Degree distribution of fixed k converges to a normal law.

Universal asymptotic behavior observed across subcritical classes.

Abstract

We present a unified general method for the asymptotic study of graphs from the so-called "subcritical"$ $ graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works both in the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number $g_n/n!$ (resp. $g_n$) of labelled (resp. unlabelled) graphs on $n$ vertices from a subcritical graph class ${G}=\cup_n {G_n}$ satisfies asymptotically the universal behaviour $$ g_n = c n^{-5/2} \gamma^n (1+o(1)) $$ for computable constants $c,\gamma$, e.g. $\gamma\approx 9.38527$ for unlabelled series-parallel graphs, and that the number of vertices of degree $k$ ($k$ fixed) in a graph chosen uniformly at random from $G_n$, converges (after rescaling) to a normal law as $n\to\infty$.