Riemann-Hurwitz theorem and second main theorem for harmonic morphisms on graphs or metrized complexes

TL;DR

This paper extends classical theorems like Riemann-Hurwitz and second main theorem to harmonic morphisms on graphs and metrized complexes, blending algebraic geometry and graph theory.

Contribution

It introduces new Riemann-Hurwitz and second main theorems for harmonic morphisms on various graph structures, inspired by recent research in the area.

Findings

Established Riemann-Hurwitz theorems for harmonic morphisms on graphs and complexes.

Developed second main theorems for harmonic morphisms in the context of Nevanlinna theory.

Unified framework for harmonic morphisms on different graph types.

Abstract

In this article, we mainly obtain the Riemann-Hurwitz theorems for harmonic morphisms on (vertex-weighted) metric graphs or metrized complexes of algebraic curves, inspired of the recent work on harmonic morphisms of graphs or metrized complexes due to many researchers. By making use of these Riemann-Hurwitz theorems, we then systematically establish the second main theorems for harmonic morphisms on finite graphs, vertex-weighted graphs, (vertex-weighted) metric graphs or metrized complexes of algebraic curves, from the viewpoint of Nevanlinna theory.