Spanning trees in random series-parallel graphs

TL;DR

This paper uses analytic techniques to estimate the expected number of spanning trees in random series-parallel graphs, providing explicit constants and extending results to various subfamilies.

Contribution

It introduces a precise asymptotic estimate for the expected number of spanning trees in random series-parallel graphs and related subfamilies, with computable constants.

Findings

Expected number of spanning trees follows an exponential decay with graph size

Constants for the asymptotic estimate are approximately s ≈ 0.09063 and ρ^{-1} ≈ 2.08415

Results extend to 2-connected series-parallel graphs, 2-trees, and fixed excess subfamilies

Abstract

By means of analytic techniques we show that the expected number of spanning trees in a connected labelled series-parallel graph on $n$ vertices chosen uniformly at random satisfies an estimate of the form $s \varrho^{-n} (1+o(1))$, where $s$ and $\varrho$ are computable constants, the values of which are approximately $s \approx 0.09063$ and $\varrho^{-1} \approx 2.08415$. We obtain analogue results for subfamilies of series-parallel graphs including 2-connected series-parallel graphs, 2-trees, and series-parallel graphs with fixed excess.