Here, we tried to summarize all results that were ever published on the topic of quantum graphs. Although the area is very young, the list already contains over 25 articles. Our hope is that our conference will help multiplying the number of results. In case we forgot some work, please contact us, so that we can fill the gap.
Runyao Duan. Super-Activation of Zero-Error Capacity of Noisy Quantum Channels. arXiv: 0906.2527
Runyao Duan, Simone Severini, and Andreas Winter. Zero-Error Communication via Quantum Channels, Noncommutative Graphs, and a Quantum Lovász Number. IEEE Transactions on Information Theory. 2013, 59(2), 1164–1174. doi:10.1109/TIT.2012.2221677
Nik Weaver. Quantum Relations. Memoirs of the American Mathematical Society. 2012, 215(1010), 81–140. doi:10.1090/S0065-9266-2011-00637-4
Dan Stahlke. Quantum Zero-Error Source-Channel Coding and Non-Commutative Graph Theory. IEEE Transactions on Information Theory. 2016, 62(1) 554–577 doi:10.1109/TIT.2015.2496377.
Nik Weaver. Quantum Graphs as Quantum Relations. The Journal of Geometric Analysis. 2021, 31 9090–9112. doi:10.1007/s12220-020-00578-w
Benjamin Musto, David Reutter, and Dominic Verdon. A compositional approach to quantum functions. Journal of Mathematical Physics. 2018, 59(8), 081706. doi:10.1063/1.5020566
Benjamin Musto, David Reutter, and Dominic Verdon. The Morita Theory of Quantum Graph Isomorphisms. Communications in Mathematical Physics. 2019, 365 797–845. doi:10.1007/s00220-018-3225-6
Michael Brannan, Alexandru Chirvasitu, Kari Eifler, Samuel Harris, Vern Paulsen, Xiaoyu Su, and Mateusz Wasilewski. Bigalois Extensions and the Graph Isomorphism Game. Communications in Mathematical Physics. 2020, 375 1177–1809. doi:10.1007/s00220-019-03563-9
Javier Alejandro Chávez-Domínguez, Andrew T. Swift. Connectivity for quantum graphs. Linear Algebra and its Applications. 2021, 608 37-53. doi:10.1016/j.laa.2020.08.020
Gareth Boreland, Ivan G. Todorov, Andreas Winter. Sandwich theorems and capacity bounds for non-commutative graphs. Journal of Combinatorial Theory, Series A. 2021, 177, 105302. doi:10.1016/j.jcta.2020.105302
Michael Brannan, Priyanga Ganesan, and Samuel J. Harris. The quantum-to-classical graph homomorphism game. Journal of Mathematical Phyiscs. 2022, 63 112204. doi:10.1063/5.0072288
Michael Brannan, Kari Eifler, Christian Voigt, and Moritz Weber. Quantum Cuntz–Krieger algebras. Transactions of the American Mathematical Society Series B. 2022 9 782–826. doi:10.1090/btran/88
Gareth Boreland, Ivan G. Todorov, Andreas Winter. Information theoretic parameters of noncommutative graphs and convex corners. Illinois Journal of Mathematics. 2022, 66(2): 123–187. doi:10.1215/00192082-9799163
Alexandru Chirvasitu and Mateusz Wasilewski. Random quantum graphs. Transactions of the American Mathematical Society. 2022, 375 3061–3087. doi:10.1090/tran/8584
Michael Brannan, Samuel J. Harris, Ivan G. Todorov, and Lyudmila Turowska. Synchronicity for quantum non-local games. Journal of Functional Analysis. 2023, 284(2), 109738. doi:10.1016/j.jfa.2022.109738
Michael Brannan, Mitch Hamidi, Lara Ismert, Brent Nelson, Mateusz Wasilewski. Quantum edge correspondences and quantum Cuntz-Krieger algebras. Journal of the London Mathematical Society. 2022, 3 886-913. doi:10.1112/jlms.12702
Priyanga Ganesan. Spectral bounds for the quantum chromatic number of quantum graphs. Linear Algebra and its Applications. 2023, 674 351–376. doi:10.1016/j.laa.2023.06.007
Daniel Gromada. Some examples of quantum graphs. Letters in Mathematical Physics. 2022, 112 122. doi:10.1007/s11005-022-01603-5
Junichiro Matsuda. Classification of quantum graphs on 𝑀2 and their quantum automorphism groups. Journal of Mathematical Physics. 2022, 63 (9), 092201. doi:10.1063/5.0081059
Matthew Daws. Quantum graphs: Different perspectives, homomorphisms and quantum automorphisms. Communications of the American Mathematical Society. 2024, 4 117–181. doi:10.1090/cams/30
Daniel Gromada. Quantum symmetries of Hadamard matrices. Transactions of the American Mathematical Society. 2024, 377(9) 6341–6377. doi:10.1090/tran/9153
Michael Brannan, Samuel J. Harris, Ivan G. Todorov, Lyudmila Turowska. Quantum no-signalling bicorrelations. Advances in Mathematics. 2024, 449 109732. doi:10.1016/j.aim.2024.109732
Dominic Verdon. Covariant Quantum Combinatorics with Applications to Zero-Error Communication. Communincations in Mathematical Physics. 2024, 405, 51. doi:10.1007/s00220-023-04898-0
Arkadiusz Bochniak, and Paweł Kasprzak. Quantum Mycielski Graphs. arXiv:2306.09994
Mateusz Wasilewski. On quantum Cayley graphs. Documenta Mathematica. 2024, 29(6) 1281–1317. doi:10.4171/DM/987
Eric Culf, and Arthur Mehta. New Approaches to Complexity via Quantum Graphs. arXiv:2309.12887
Junichiro Matsuda. Algebraic connectedness and bipartiteness of quantum graphs. Communincations in Mathematical Physics. 2024, 405, 185. doi:10.1007/s00220-024-05046-y
Gage Hoefer, Ivan G. Todorov. Homomorphisms of quantum hypergraphs. arXiv:2311.06355
Amaury Freslon, and Moritz Weber. Discrete quantum subgroups of free unitary quantum groups. arXiv:2403.01526
Daniel Gromada. Quantum association schemes. arXiv:2404.06157
Adina Goldberg. Quantum Games and Synchronicity. arXiv:2408.15444
Wojciech Paupa, Piotr M. Sołtan. Invariants of the quantum graph of the partial trace. arXiv:2408.16858
Michael Brannan, Roberto Hernández Palomares. Quantum graphs, subfactors and tensor categories I. arXiv:2409.01951
Michael Brannan, Eric Culf, and Matthijs Vernooij. Quantum expanders and property (T) discrete quantum groups. arXiv:2502.01974
Mateusz Wasilewski. Lecture Notes on Quantum Graphs