TITLE: Generating function in graph theory
QUESTION [19 upvotes]: I am looking for a simple illustration of generating functions in graph theory.
So far, the matching polynomial seems to be the best. But I want something bit richer; at least a derivative should show up.
Please help.

REPLY [3 votes]: Generating functions are ubiquitous in the enumerative theory of maps, that is, graphs on surfaces. For planar graphs this theory was first developed in the seminal work of Tutte in the late 60s through early 80s. But in recent years there has also been a lot of work devoted to the higher genus maps as well. It turns out that the generating functions for these objects satisfy sophisticated differential equations (coming from the "KP hierary"); see for instance Goulden and Jackson, "The KP hierarchy, branched covers, and triangulations" (https://arxiv.org/abs/0803.3980) or Carrell and Chaupy, "A simple recurrence formula for the number of rooted maps on surfaces by edges and genus" (https://dmtcs.episciences.org/2424/)