Graph Theory and Its Applications in Quantum Mechanics

Dear Colleagues,

Over recent decades, there has been a growing interest in the interplay between quantum mechanics and graph theory. This rich and diverse area encompasses topics ranging from the modeling of nanoscale devices, such as spin chains, to the understanding of natural processes like energy transfer in biological systems. Moreover, quantum networks and devices governed by the laws of quantum mechanics are becoming increasingly feasible, offering new opportunities for both fundamental studies and technological applications.

Numerous concepts from graph theory have been successfully applied to quantum systems, including the study of correlations, entanglement structures, transport phenomena, and the dynamics of composite systems. In many cases, these systems are naturally represented as graphs, and quantum dynamics on graphs (which can be simulated efficiently even on standard personal computers) offer valuable insights into both the physical behavior of the system and the mathematical properties of the underlying graphs.

Several research directions illustrate this vibrant interface. Graph states form a class of multipartite entangled states associated with graphs, encompassing paradigmatic examples such as GHZ and cluster states. Quantum walks, describing the evolution of quantum particles on graphs, have become important tools in quantum computing and algorithm development. Quantum graphs, originally inspired by the Kronig–Penney model, have been widely applied to study transport phenomena, scattering, wave propagation, and quantum chaos. Furthermore, quantum graphs have recently found applications as effective models for wave filtering devices and metamaterials, where their topological and spectral properties can be engineered to achieve specific control over wave transmission and localization. Additionally, spin chains continue to be intensively investigated as promising platforms for quantum state transfer, with graph-theoretic methods playing a crucial role in optimizing their performance.

This Special Issue seeks to highlight recent developments at the interface of graph theory and quantum mechanics and invites submissions of original research articles as well as comprehensive reviews. Topics of interest include, but are not limited to, the following:

• Quantum graphs and wave dynamics in network-like structures;
• Quantum walks and their computational applications;
• Graph states and multipartite entanglement ;
• Quantum state transfer in spin chains and graph-based architectures;
• Quantum communication and network routing on graphs;
• Applications of graph theory to quantum chaos, filtering, and metamaterials.

Dr. Fabiano Manoel de Andrade
Guest Editor

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