Spectral comparison results for Laplacians on discrete graphs

In the recent literature, various authors have studied spectral comparison results for Schrödinger operators with discrete spectrum in different settings including Euclidean domains and quantum graphs. In this note, we derive such spectral comparison results in a rather general framework for general and possibly infinite discrete graphs. Along the way, we establish a discrete version of the local Weyl law whose proof does neither involve any Tauberian theorem nor the Weyl law as used in the continuous case.