The adjacency matrix and the discrete Laplacian acting on forms

TL;DR

This paper explores the relationship between the adjacency matrix and the discrete Laplacian on 1-forms, analyzing conditions for essential self-adjointness and discussing the concept of completeness in this context.

Contribution

It provides new insights into the connection between adjacency matrices and discrete Laplacians on forms, especially regarding self-adjointness properties.

Findings

Boundedness from below does not imply essential self-adjointness.

Conditions for essential self-adjointness are examined.

Discussion on the notion of completeness in this setting.

Abstract

We study the relationship between the adjacency matrix and the discrete Laplacian acting on 1-forms. We also prove that if the adjacency matrix is bounded from below it is not necessarily essentially self-adjoint. We discuss the question of essential self-adjointness and the notion of completeness.