Random graphs with forbidden vertex degrees

We study the random graph Gn,λ/n conditioned on the event that all vertex degrees lie in some given subset S of the nonnegative integers. Subject to a certain hypothesis on S, the empirical distribution of the vertex degrees is asymptotically Poisson with some parameter µ given as the root of a certain ‘characteristic equation’ of S that maximizes a certain function ψS (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.

• Type: Journal
• Year: 2010
• Where: RSA
• Authors: Geoffrey R. Grimmett, Svante Janson
• Keywords: Certain ‘characteristic Equation, Random Graph, RSA 2010, Vertex Degrees