  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.
