Unboundedness of adjacency matrices of locally finite graphs

Abstract

Given a locally finite simple graph so that its degree is not bounded, every self-adjoint realization of the adjacency matrix is unbounded from above. In this note we give an optimal condition to ensure it is also unbounded from below. We also consider the case of weighted graphs. We discuss the question of self-adjoint extensions and prove an optimal criterium.

Typos corrected. Examples added. Cute drawings. Simplification of the main condition. Case of the weight tending to zero more discussed.

References in corpus (1)

• Essential selfadjointness of the graph-Laplacian

Cited by in corpus (3)

• The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs
• Essential self-adjointness of magnetic Schrödinger operators on locally finite graphs
• The Essential Spectrum of the Discrete Laplacian on Klaus-sparse Graphs