Enumeration of connected bipartite graphs with given Betti number

TL;DR

This paper derives differential equations for generating functions counting connected bipartite graphs with a fixed Betti number, solves them explicitly, and analyzes their asymptotic behavior, introducing a classification via basic graphs.

Contribution

It introduces a novel differential equation approach to enumerate connected bipartite graphs with a given Betti number and provides explicit formulas and asymptotics.

Findings

Derived differential equations for generating functions

Explicit formulas for counting graphs with fixed Betti number

Asymptotic analysis of graph enumeration results

Abstract

We obtain first order linear partial differential equations which are satisfied by exponential generating functions of two variables for the number of connected bipartite graphs with given Betti number. By solving these equations inductively, we obtain the explicit form of generating functions and derive the asymptotic behavior of their coefficients. We also introduce a family of basic graphs to classify connected bipartite graphs and give another expression of the generating functions as the sum over basic graphs of rational functions of those for the number of labeled bipartite rooted spanning trees.