A note on the Gaffney Laplacian on infinite metric graphs

TL;DR

This paper investigates the properties of the Gaffney Laplacian on infinite metric graphs, establishing a relationship between deficiency indices and graph ends, and providing criteria for the operator's closure based on these ends.

Contribution

It introduces new criteria linking finite volume graph ends to the closure of the Gaffney Laplacian on infinite metric graphs.

Findings

Deficiency indices equal the number of finite volume graph ends.

Criteria for the Gaffney Laplacian to be closed are provided.

Relationship between graph topology and Laplacian properties established.

Abstract

We show that the deficiency indices of the minimal Gaffney Laplacian on an infinite locally finite metric graph are equal to the number of finite volume graph ends. Moreover, we provide criteria, formulated in terms of finite volume graph ends, for the Gaffney Laplacian to be closed.