Connectivity for Quantum Graphs via Quantum Adjacency Operators

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 Kubo-Martin-Schwinger (KMS) inner product, which unifies and generalizes existing formulations. We also prove that quantum graph connectivity is preserved by quantum isomorphisms.