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.
Highlights
Introduction and Main ResultsA graph G = (V (G), E(G)) is permitted to have both loops and multiple edges
A embedding of a graph G into an orientable surface Ok is a cellular embedding, i.e., the interior of every face is homeomorphic to an open disc
We denote the number of cellular embeddings of G on the surface Ok by gk(G), where, by the number of embeddings, we mean the number of equivalence classes under ambient isotopy
Summary
The average genus γavg(G) of the graph G is the expected value of the genus random variable, over all labeled 2-cell orientable embeddings of G, using the uniform distribution. One objective of this paper is to give an explicit expression of the average genus for a bouquet of circles. In this paper, using knowledge in ordinary differential equations and Taylor formulas, we derive an explicit expression of γavg(Bn). Another objective of this paper is to give an explicit expression of the average genus for dipoles Dn (two vertices, n−multiple edges). Our method is new and deeply depends on the knowledge in ordinary differential equations and series theory
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