Asymptotic expansion of regular and connected regular graphs

TL;DR

This paper derives detailed asymptotic expansions for k-regular and connected k-regular graphs, providing precise error estimates and connecting the expansions through classical divergent series techniques.

Contribution

It introduces a novel application of the Laplace method to derive asymptotics for regular graphs and extends the analysis to connected graphs using Wright and Bender's techniques.

Findings

Asymptotic expansion of k-regular graphs with arbitrary error terms

Asymptotic expansion of connected k-regular graphs

Quantitative comparison between the two expansions

Abstract

We derive the asymptotic expansion (asymptotics with an arbitrary number of error terms) of k-regular graphs by applying the Laplace method on a recent exact formula from Caizergues and de Panafieu (2023). We also deduce the asymptotic expansion of connected k-regular graphs using standard techniques for divergent series developed by Wright (1970) and Bender (1975), and quantify its closeness to the asymptotic expansion of k-regular graphs.