# Random cographs: Brownian graphon limit and asymptotic degree distribution

## TL;DR

This paper studies large random cographs, showing they converge to a Brownian graphon, with degrees and connectivity properties analyzed, revealing differences between labeled and unlabeled cases.

## Contribution

It establishes the Brownian graphon limit for random cographs and analyzes degree distribution and connectivity, highlighting differences between labeled and unlabeled graphs.

## Key findings

- Convergence to a Brownian graphon in large cographs.
- Degree of a random vertex converges to Lebesgue measure after normalization.
- Connectivity distribution converges without renormalization, differing in labeled and unlabeled cases.

## Abstract

We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex in a uniform cograph is of order $n$, and converges after normalization to the Lebesgue measure on $[0,1]$. We finally analyze the vertex connectivity (i.e. the minimal number of vertices whose removal disconnects the graph) of random connected cographs, and show that this statistics converges in distribution without renormalization. Unlike for the graphon limit and for the degree of a random vertex, the limiting distribution is different in the labeled and unlabeled settings. Our proofs rely on the classical encoding of cographs via cotrees. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.