The surjectivity of the combinatorial Laplacian on infinite graphs

Abstract

Given a connected locally finite simplicial graph $G$ with vertex set $V$, the combinatorial Laplacian ${\Delta }_G:{\mathbb{R}}^V\to {\mathbb{R}}^V$ is defined on the space of all real-valued functions on $V$. We prove that ${\Delta }_G$ is surjective if $G$ is infinite.