Asymptotic expansion of regular and connected regular graphs

Élie de Panafieu

Published 2024-08-22Version 1

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). Previously, only the main asymptotics was known for general k and the asymptotic expansions were known only up to k=7 (Chyzak, Mishna (2024)). 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.