Random graphs with forbidden vertex degrees

Geoffrey Grimmett; Svante Janson

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arXiv · 0712.0270

Random graphs with forbidden vertex degrees

Geoffrey Grimmett·Svante Janson

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Abstract

We study the random graph G_{n,λ/n} conditioned on the event that all vertex degrees lie in some given subset S of the non-negative integers. Subject to a certain hypothesis on S, the empirical distribution of the vertex degrees is asymptotically Poisson with some parameter \mux given as the root of a certain `characteristic equation' of S that maximises a certain function \psis(μ). Subject to a hypothesis on S, we obtain a partial description of the structure of such a random graph, including a condition for the existence (or not) of a giant component. The requisite hypothesis is in many cases benign, and applications are presented to a number of choices for the set S including the sets of (respectively) even and odd numbers. The random \emph{even} graph is related to the random-cluster model on the complete graph K_n.

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Geoffrey Grimmett, Svante Janson. 2007-12-03. Random graphs with forbidden vertex degrees. https://arxiv.org/abs/0712.0270

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