The surjectivity of the combinatorial Laplacian on infinite graphs

Given a connected locally finite simplicial graph $G$ with vertex set $V$, the combinatorial Laplacian ${\mathrm{\Delta }}_G\text{\hspace{0.17em}⁣}:{\mathbb{R}}^V\to {\mathbb{R}}^V$ is defined on the space of all real-valued functions on $V$. We prove that ${\mathrm{\Delta }}_G$ is surjective if $G$ is infinite.