Some Enumerative Results on Series-Parallel Networks

Abstract

This chapter provides some enumerative results on series-parallel networks. The class of series-parallel networks is defined as the graph consisting of a single edge joining two terminals; the trivial network and any graph formed by joining two not necessarily distinct series-parallel networks in series or in parallel is also a series-parallel network. If a series-parallel network has one or more cut-nodes, then it is called a series network or a σ-network; if not, it is called a parallel-network or a π-network. Two π -networks or two σ-networks are regarded as the same if and only if they have the same constituents; thus the ordering of the constituents and their terminals is not taken into account. In addition, the chapter derives relations for the generating functions for various numbers. To determine the asymptotic behavior of the coefficients in these generating functions, the chapter appeals to the cases of a result of Darboux.