Unboundedness of adjacency matrices of locally finite graphs

TL;DR

This paper investigates the spectral properties of adjacency matrices of locally finite graphs, establishing conditions under which these matrices are unbounded both above and below, and explores self-adjoint extensions.

Contribution

It provides an optimal criterion for the unboundedness from below of adjacency matrices and analyzes self-adjoint extensions for weighted graphs.

Findings

Adjacency matrices of graphs with unbounded degree are unbounded from above.

An optimal condition for unboundedness from below is established.

Criteria for self-adjoint extensions of weighted graph adjacency matrices are proved.

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.