A little over 10 years ago, M. Baker and S. Norine (I’ve also seen this name spelled Norin) wrote a terrific paper on harmonic morphisms between simple, connected graphs (see “Harmonic morphisms and hyperelliptic graphs” – you can find a downloadable pdf on the internet of you google for it). Roughly speaking, a harmonic function on a graph is a function in the kernel of the graph Laplacian. A harmonic morphism between graphs is, roughly speaking, a map from one graph to another that preserves harmonic functions.
They proved quite a few interesting results but one of the most interesting, I think, is their graph-theoretic analog of the Riemann-Hurwitz formula. We define the genus of a simple connected graph 
It represents the minimum number of edges that must be removed from the graph to make it into a tree (so, a tree has genus 0).
Riemann-Hurwitz formula (Baker and Norine): Let 
![{\rm genus}(\Gamma_2)-1 = {\rm deg}(\phi)({\rm genus}(\Gamma_1)-1)+\sum_{x\in V_2} [m_\phi(x)+\frac{1}{2}\nu_\phi(x)-1].](https://s0.wp.com/latex.php?latex=%7B%5Crm+genus%7D%28%5CGamma_2%29-1+%3D+%7B%5Crm+deg%7D%28%5Cphi%29%28%7B%5Crm+genus%7D%28%5CGamma_1%29-1%29%2B%5Csum_%7Bx%5Cin+V_2%7D+%5Bm_%5Cphi%28x%29%2B%5Cfrac%7B1%7D%7B2%7D%5Cnu_%5Cphi%28x%29-1%5D.&bg=ffffff&fg=323232&s=0&c=20201002)
I’m not going to define them here but 
I simply want to record a very easy corollary to this, assuming 
Corollary: Let 
to a connected
Then
Yet Another Mathblog 