A short remark on the surjectivity of the combinatorial Laplacian on infinite graphs

Abstract

Applying a well-known theorem due to Eidelheit, we give a short proof of the surjectivity of the combinatorial Laplacian on connected locally finite undirected simplicial graph $G$ with countably infinite vertex set $V$, established by Ceccherini-Silberstein, Coornaert, and Dodziuk. In fact, we show that every linear operator on $\mathbb{K}^V$ which has finite hopping range and satisfies the pointwise maximum principle is surjective.