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

TL;DR

This paper provides a concise proof that the combinatorial Laplacian on certain infinite graphs is surjective, extending previous results by applying a classical theorem and broadening the class of operators known to be surjective.

Contribution

It offers a new, simplified proof of Laplacian surjectivity on infinite graphs and generalizes the result to operators with finite hopping range satisfying the maximum principle.

Findings

The combinatorial Laplacian is surjective on connected locally finite infinite graphs.

Operators with finite hopping range satisfying the maximum principle are surjective.

The proof leverages Eidelheit's theorem for a streamlined argument.

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.