圆和偶极子束的平均亏格
The average genus for bouquets of circles and dipoles
摘要：
圈簇$B_n$和偶极图$D_n$是拓扑图论中两类重要的图。对于$n\geq 1$，我们给出了$B_n$的平均亏格$γ_{avg}(B_n)$的显式表达式.通过该表达式，可以容易地看到$γ_{avg}(B_n)=\frac{n-\ln n-c+1-\ln 2}{2}+o(1)$，其中$c$是欧拉常数。对于$D_n$也获得了类似的结果。我们的方法是新的，深深地依赖于常微分方程的知识。
英文摘要：
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 $γ_{avg}(B_n)$ of $B_n$. By this expression, one easily sees $γ_{avg}(B_n)=\frac{n-\ln n-c+1-\ln 2}{2}+o(1)$, 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.
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