The surjectivity of the combinatorial Laplacian on infinite graphs
  Given a connected locally finite simplicial graph $ G$ with vertex set $V$, the combinatorial Laplacian $\Delta_G \colon \R^V \to \R^V$ is defined on the space of all real-valued functions on $V$. We prove that $\Delta_G$ is surjective if $G$ is infinite. 
Tullio Ceccherini-Silberstein, Michel Coornaert, and Jozef Dodziuk
math.AP math.CO