Asymptotic expansions for enumerating connected labelled graphs

I compute several terms of the asymptotic expansion of the number of connected labelled graphs with n nodes and m edges, for small k=m-n.

💡 Research Summary

The paper addresses the long‑standing combinatorial problem of counting connected labelled graphs with a given number of vertices n and edges m, focusing on the regime where the excess k = m − n is small. Starting from the classical generating‑function framework, the author defines the bivariate exponential generating function

 G(z,w)=∑{n,m}c{n,m},z^{n}w^{m}/n!

where c_{n,m} denotes the number of connected labelled graphs on n vertices with m edges. By taking the logarithm, G(z,w)=exp(C(z,w)), the function C(z,w) isolates the contribution of connected graphs. The key technical step is to expand C(z,1+u) as a power series in u, because the coefficient of u^{k} directly yields c_{n,n+k}. Using Lagrange inversion, Stirling’s approximation for factorials, and Bernoulli numbers, the author derives an explicit asymptotic expansion of the form

 c_{n,n+k}=n^{n‑2}\frac{n^{k}}{k!}\Bigl