Vertices of given degree in series-parallel graphs

We show that the number of vertices of a given degree k in several kinds of series-parallel labelled graphs of size n satisfy a central limit theorem with mean and variance proportional to n, and quadratic exponential tail estimates. We further prove a corresponding theorem for the number of nodes of degree two in labelled planar graphs. The proof method is based on generating functions and singularity analysis. In particular we need systems of equations for multivariate generating functions and transfer results for singular representations of analytic functions.

• Type: Journal
• Year: 2010
• Where: RSA
• Authors: Michael Drmota, Omer Giménez, Marc Noy
• Keywords: Multivariate Generating Functions, Planar Graphs, Quadratic Exponential Tail, RSA 2010