Connectivity for quantum graphs via quantum adjacency operators

TL;DR

This paper develops an algebraic framework to characterize connectivity in quantum graphs using quantum adjacency matrices, extending previous models and employing a quantum Perron-Frobenius theorem.

Contribution

It introduces a spectral characterization of connectivity for general quantum graphs, including non-tracial and non-regular cases, via the KMS inner product.

Findings

Connectivity characterized by spectral properties of quantum adjacency matrices

Irreducibility of the quantum adjacency matrix implies connectivity

Nullity of the quantum graph Laplacian relates to connectivity

Abstract

Connectivity is a fundamental property of quantum graphs, previously studied in the operator system model for matrix quantum graphs and via graph homomorphisms in the quantum adjacency matrix model. In this paper, we develop an algebraic characterization of connectivity for general quantum graphs within the quantum adjacency matrix framework. Our approach extends earlier results to the non-tracial setting and beyond regular quantum graphs. We utilize a quantum Perron-Frobenius theorem that provides a spectral characterization of connectivity, and we further characterize connectivity in terms of the irreducibility of the quantum adjacency matrix and the nullity of the associated graph Laplacian. These results are obtained using the KMS inner product, which unifies and generalizes existing formulations.