Spectral multiplicity functions of adjacency operators of graphs and cospectral infinite graphs

TL;DR

This paper reviews spectral multiplicity functions of adjacency operators in infinite graphs and provides explicit examples of cospectral and uniquely determined graphs, enhancing understanding of spectral graph theory.

Contribution

It offers a systematic review of spectral multiplicity functions and constructs explicit examples of cospectral and uniquely determined infinite graphs.

Findings

Identified examples of infinite graphs with known spectral multiplicity functions

Constructed explicit cospectral infinite graphs

Presented examples of graphs uniquely determined by their spectra

Abstract

The adjacency operator of a graph has a spectrum and a class of scalar-valued spectral measures which have been systematically analyzed; it also has a spectral multiplicity function which has been less studied. The first purpose of this article is to review some examples of infinite graphs for which the spectral multiplicity function of the adjacency operator has been determined. The second purpose of this article is to show explicit examples of infinite connected graphs which are cospectral, i.e., which have unitarily equivalent adjacency operators, and explicit examples of infinite connected graphs which are uniquely determined by their spectrum.