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L’Enseignement Mathématique

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Volume 58, Issue 1/2, 2012, pp. 125–130
DOI: 10.4171/LEM/58-1-5

Published online: 2012-06-30

The surjectivity of the combinatorial Laplacian on infinite graphs

Tullio Ceccherini-Silberstein[1], Michel Coornaert[2] and Jozef Dodziuk[3]

(1) Università del Sannio, Benevento, Italy
(2) Université de Strasbourg, France
(3) The CUNY Graduate Center, New York, United States

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

Keywords: simplicial graph, combinatorial Laplacian, Mittag-Leffler lemma, maximum principle, surjectivity

Ceccherini-Silberstein Tullio, Coornaert Michel, Dodziuk Jozef: The surjectivity of the combinatorial Laplacian on infinite graphs. Enseign. Math. 58 (2012), 125-130. doi: 10.4171/LEM/58-1-5