Institute of
Discrete Mathematics and Geometry

We study the distribution of node degrees in subcritical families of graphs. These are families which are block-stable, that is they are determined only by their 2-connected components, and the generating function of 2-connected components fulfills certain analytic conditions, which we call subcriticality conditions. Important subcritical families are outerplanar and series-parallel graphs. We extend recent results by Drmota, Gimenez and Noy in the labelled case, focusing mainly on the unlabelled framework. We use a generating functions approach, taking into account symmetries evolving from unlabelling and therefore using Polya's theory on cycle index sums. By methods of singularity analysis we can prove a central limit law for the number of nodes of given degree k in a random unlabelled graph of size n in any subcritical family of graphs. Therefore we obtain a degree distribution. This results holds generally in the subcritical framework, without any further knowledge on the structure of 2-connected components.