The average genus for bouquets of circles and dipoles

Jinlian Zhang, Xuhui Peng, Yichao Chen

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.


Keywords


Average genus; Bouquet of circles; Dipole; Ordinary differential equations

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DOI: https://doi.org/10.26493/1855-3974.2043.fbb

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications