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Quantum isomorphic strongly regular graphs from the E 8 root system
Schmidt, Simon1
1 QMATH Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 2100 Copenhagen Ø Denmark
Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 515-528.

In this article, we give a first example of a pair of quantum isomorphic, non-isomorphic strongly regular graphs, that is, non-isomorphic strongly regular graphs having the same homomorphism counts from all planar graphs. The pair consists of the orthogonality graph of the 120 lines spanned by the E 8 root system and a rank 4 graph whose complement was first discovered by Brouwer, Ivanov and Klin. Both graphs are strongly regular with parameters (120,63,30,36). Using Godsil-McKay switching, we obtain more quantum isomorphic, non-isomorphic strongly regular graphs with the same parameters.

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Accepted:
Published online:
DOI: 10.5802/alco.335
Classification: 05E30, 20B25, 05C25, 17B22, 20G42
Keywords: strongly regular graphs, quantum isomorphism, root systems, Godsil–McKay switching

Schmidt, Simon 1

1 QMATH Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 2100 Copenhagen Ø Denmark
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Schmidt, Simon. Quantum isomorphic strongly regular graphs from the $E_8$ root system. Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 515-528. doi : 10.5802/alco.335. https://alco.centre-mersenne.org/articles/10.5802/alco.335/

[1] Atserias, Albert; Mančinska, Laura; Roberson, David E.; Šámal, Robert; Severini, Simone; Varvitsiotis, Antonios Quantum and non-signalling graph isomorphisms, J. Combin. Theory Ser. B, Volume 136 (2019), pp. 289-328 | DOI | MR

[2] Brannan, Michael; Chirvasitu, Alexandru; Eifler, Kari; Harris, Samuel; Paulsen, Vern; Su, Xiaoyu; Wasilewski, Mateusz Bigalois extensions and the graph isomorphism game, Comm. Math. Phys., Volume 375 (2020) no. 3, pp. 1777-1809 | DOI | MR

[3] Brouwer, A. E.; Ivanov, A. V.; Klin, M. H. Some new strongly regular graphs, Combinatorica, Volume 9 (1989) no. 4, pp. 339-344 | DOI | MR

[4] Brouwer, Andries E.; Van Maldeghem, H. Strongly regular graphs, Encyclopedia of Mathematics and its Applications, 182, Cambridge University Press, Cambridge, 2022, xvii+462 pages | DOI

[5] Chan, Ada; Martin, William J. Quantum isomorphism of graphs from association schemes, J. Combin. Theory Ser. B, Volume 164 (2024), pp. 340-363 | DOI | MR

[6] Godsil, C. D.; McKay, B. D. Constructing cospectral graphs, Aequationes Math., Volume 25 (1982) no. 2-3, pp. 257-268 | DOI | MR

[7] Gromada, Daniel Quantum symmetries of Hadamard matrices, 2022 | arXiv

[8] Lupini, Martino; Mančinska, Laura; Roberson, David E. Nonlocal games and quantum permutation groups, J. Funct. Anal., Volume 279 (2020) no. 5, Paper no. 108592, 44 pages | MR

[9] Mančinska, Laura; Roberson, David E. Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), IEEE (2020), pp. 661-672

[10] Mathon, Rudolf; Street, Anne Penfold Overlarge sets and partial geometries, J. Geom., Volume 60 (1997) no. 1-2, pp. 85-104 | DOI | MR

[11] Musto, Benjamin; Reutter, David; Verdon, Dominic The Morita theory of quantum graph isomorphisms, Comm. Math. Phys., Volume 365 (2019) no. 2, pp. 797-845 | DOI | MR | Zbl

[12] Roberson, David E.; Schmidt, Simon Solution group representations as quantum symmetries of graphs, J. Lond. Math. Soc. (2), Volume 106 (2022) no. 4, pp. 3379-3410 | DOI | MR

[13] Schmidt, Simon Quantum automorphism groups of finite graphs, Ph. D. Thesis, Saarländische Universitäts-und Landesbibliothek (2020)

[14] The Sage Developers SageMath, the Sage Mathematics Software System (Version 8.9) (2019) (https://www.sagemath.org)

[15] Wang, Shuzhou Quantum symmetry groups of finite spaces, Comm. Math. Phys., Volume 195 (1998) no. 1, pp. 195-211 | DOI | MR | Zbl

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