The surjectivity of the combinatorial Laplacian on infinite graphs

TL;DR

This paper proves that the combinatorial Laplacian operator on an infinite, connected, locally finite graph is surjective, extending understanding of Laplacian properties in infinite graph theory.

Contribution

It establishes the surjectivity of the combinatorial Laplacian on infinite graphs, a property not previously confirmed for such structures.

Findings

The Laplacian is surjective on infinite graphs.

Surjectivity holds for connected, locally finite, infinite graphs.

This result advances the theoretical understanding of Laplacians in infinite graph contexts.

Abstract

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.