The average genus for bouquets of circles and dipoles

TL;DR

This paper derives explicit formulas for the average genus of bouquets of circles and dipole graphs, revealing asymptotic behavior and introducing a novel method based on differential equations.

Contribution

It provides the first explicit formulas for the average genus of these graph classes using a new approach involving ordinary differential equations.

Findings

Average genus of $B_n$ approximates to (n - ln n - c + 1 - ln 2)/2

Similar formulas obtained for $D_n$

Method introduces a novel differential equations approach

Abstract

The bouquet of circles $B_n$ and dipole graph $D_n$ are two important classes of graphs in topological graph theory. For $n\geq 1$, we give an explicit formula for the average genus $\gamma_{avg}(B_n)$ of $B_n$. By this expression, one easily sees $\gamma_{avg}(B_n)=\frac{n-\ln n-c+1-\ln 2}{2}+o(1)$, where $c$ is the Euler constant. Similar results are obtained for $D_n$. Our method is new and deeply depends on the knowledge in ordinary differential equations.