Airy Phenomena and Analytic Combinatorics of Connected Graphs
Résumé
The enumeration of connected graphs has been so far dealt with by probabilistic methods, by special combinatorial decompositions or by somewhat indirect formal series manipulations. We show here that it is possible to make analytic sense of the divergent series that expresses the generating function of connected graphs. As a consequence, it becomes possible to derive analytically known enumeration results using only first principles of combinatorial analysis and straight asymptotic analysis---specifically, the saddle-point method. The enumeration of connected graphs by excess derives from a simple saddle-point analysis; furthermore, a refined analysis based on coalescent saddle points yields an explicit connection with Airy functions.