In this paper, we try to answer some questions raised by Cangelmi (Eur J Comb 33(7):1444–1448, 2012). We reinterpret the Riemann–Hurwitz theorem of orientable algebraic hypermaps by introducing tripartite graph morphisms and obtain Riemann–Roch theorems for orientable hypermaps by defining the divisor of a function f on darts. In addition, we extend Riemann–Roch theorem to non-orientable hypermaps by suitably replacing the orientable genus with the non-orientable genus. Finally, as an application of the Riemann–Hurwitz theorem, we establish the second main theorem from the viewpoint of Nevanlinna theory.

中文翻译：

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超图的黎曼-赫尔维茨定理和黎曼-罗赫定理

在本文中，我们尝试回答 Cangelmi 提出的一些问题（Eur J Comb 33(7):1444–1448, 2012）。我们通过引入三方图态射重新解释了可定向代数超图的黎曼-赫尔维茨定理，并通过在飞镖上定义函数f的除数来获得可定向超图的黎曼-罗赫定理。此外，我们通过用不可定向亏格适当替换可定向亏格，将黎曼-罗赫定理扩展到不可定向超图。最后，作为黎曼-赫尔维茨定理的应用，我们从内万林纳理论的角度建立了第二个主定理。