The surjectivity of the combinatorial Laplacian on infinite graphs

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

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