Vertices of given degree in series-parallel graphs

We show that the numbers of vertices of a given degree k ≥ 1 in several kinds of series-parallel labeled 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 labeled 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. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010

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