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

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.