Harmonic Morphisms of Arithmetical Structures on Graphs

Let $\phi \text{\hspace{0.17em}⁣}:{\mathrm{\Gamma }}_2\to {\mathrm{\Gamma }}_1$ be a harmonic morphism of connected graphs. We show that an arithmetical structure on ${\mathrm{\Gamma }}_1$ can be pulled back via $\phi$ to an arithmetical structure on ${\mathrm{\Gamma }}_2$. We then show that some results of Baker and Norine on the critical groups for the usual Laplacian extend to arithmetical critical groups, which are abelian groups determined by the generalized Laplacian associated to these arithmetical structures. In particular, we show that the morphism $\phi$ induces a surjective group homomorphism from the arithmetical critical group of ${\mathrm{\Gamma }}_2$ to that of ${\mathrm{\Gamma }}_1$ and an injective group homomorphism from the arithmetical critical group of ${\mathrm{\Gamma }}_1$ to that of ${\mathrm{\Gamma }}_2$. Finally, we prove a Riemann-Hurwitz formula for arithmetical structures.