Connectivity for quantum graphs via quantum adjacency operators
Kristin Courtney, Priyanga Ganesan, Mateusz Wasilewski
TL;DR
The paper develops an algebraic notion of connectivity for quantum graphs within the quantum adjacency matrix framework, extending connectivity to non-tracial and non-regular settings via the KMS inner product and a quantum Perron-Frobenius theorem. It shows connectivity is equivalent to irreducibility of the quantum adjacency matrix and to the nullity properties of the quantum graph Laplacian, unifying approaches from operator systems and quantum relations. The authors extend bipartite-graph criteria to non-regular/non-tracial cases and prove an exact operator-norm result for $d$-regular quantum graphs, establishing $\|A\|=d$ in the GNS norm. Overall, the work provides a cohesive spectral and algebraic characterization of connectivity for general quantum graphs, broadening applicability and linking classical and quantum graph notions via the KMS structure.
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.