The spectra of digraphs with Morita equivalent $C^\ast$-algebras

TL;DR

This paper investigates how certain graph spectra are affected by moves that classify graph $C^*$-algebras up to Morita equivalence, revealing which spectra are preserved and providing counterexamples for others.

Contribution

It identifies which digraph spectra remain invariant under specific moves used in classifying $C^*$-algebras, extending understanding of spectral invariance in graph theory.

Findings

Adjacency spectra of digraphs and line digraphs are preserved by some moves.

The Cuntz splice preserves the skew adjacency and Laplace spectra.

Other spectra are not preserved by the moves, with counterexamples provided.

Abstract

Eilers et al. have recently completed the geometric classification of unital graph $C^\ast$-algebras up to Morita equivalence using a set of moves on the corresponding digraphs. We explore the question of whether these moves preserve the nonzero elements of the spectrum of a finite digraph, which in this paper is allowed to have loops and parallel edges. We consider several different digraph spectra that have been studied in the literature, answering this question for the Laplace and adjacency spectra, their skew counterparts, the symmetric adjacency spectrum, the adjacency spectrum of the line digraph, the Hermitian adjacency spectrum, and the normalized Laplacian, considering in most cases two ways that these spectra can be defined in the presence of parallel edges. We show that the adjacency spectra of the digraph and line digraph are preserved by a subset of the moves, and the skew adjacency and Laplace spectra are preserved by the Cuntz splice. We give counterexamples to show that the other spectra are not preserved by the remaining moves. The same results hold if one restricts to the class of strongly connected digraphs.