We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence toward 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 . 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 of the vertex connectivity 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.

中文翻译：

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Random cographs：布朗石墨极限和渐近度分布

我们考虑大尺寸的均匀随机 cographs（标记或未标记）。我们的第一个主要结果是向石墨子空间中的布朗限制对象收敛。然后，我们证明了均匀 cograph 中均匀随机顶点的度数是n阶的，并且在归一化到 Lebesgue 测度后收敛. 我们最后分析了随机连接cograph 的顶点连通性（即移除断开图的顶点的最小数量），并表明该统计量在没有重整化的情况下收敛于分布。与 graphon 限制和随机顶点的度数不同，顶点连接的限制分布在标记和未标记设置中是不同的。我们的证明依赖于通过 cotree 对 cographs 的经典编码。然后我们主要使用组合论证，包括符号方法和奇点分析。