Automorphisms of the totally isotropic subspace inclusion graph of unitary spaces

Jianmei He; Lanxin Chen; Zhengyu Guo; Jianmei He; Lanxin Chen; Zhengyu Guo

doi:10.3934/math.2026406

AIMS Mathematics

2026, Volume 11, Issue 4: 9819-9844. doi: 10.3934/math.2026406

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Automorphisms of the totally isotropic subspace inclusion graph of unitary spaces

1.
School of Science, Shijiazhuang University, Shijiazhuang 050035, China
2.
School of Sciences, Hebei University of Science and Technology, Shijiazhuang 050018, China

Received: 06 February 2026 Revised: 20 March 2026 Accepted: 30 March 2026 Published: 13 April 2026
MSC : 05C07, 05C50, 05C69

Let $ F_{q^{2}} $ be a finite field of $ q^{2} $ elements, where $ q $ is a power of a prime and $ \mathbb{U}_{n} $ an $ n $-dimensional unitary space over $ F_{q^{2}} $. The inclusion graph of the totally isotropic subspace of $ \mathbb{U}_{n} $, written as $ \mathcal{I}n(\mathbb{U}_{n}) $, is a graph which has all totally isotropic subspaces of $ \mathbb{U}_{n} $ as its vertices and two distinct, totally isotropic subspaces $ U_{1} $, $ U_{2} $ of $ \mathbb{U}_{n} $ which are adjacent if and only if $ U_{1}\subset U_{2} $ or $ U_{2}\subset U_{1} $. In this paper, all automorphisms of $ \mathcal{I}n(\mathbb{U}_{n}) $ are determined definitely.
- automorphisms,
- totally isotropic subspace,
- unitary space,
- inclusion graph
Citation: Jianmei He, Lanxin Chen, Zhengyu Guo. Automorphisms of the totally isotropic subspace inclusion graph of unitary spaces[J]. AIMS Mathematics, 2026, 11(4): 9819-9844. doi: 10.3934/math.2026406

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Abstract

Let $ F_{q^{2}} $ be a finite field of $ q^{2} $ elements, where $ q $ is a power of a prime and $ \mathbb{U}_{n} $ an $ n $-dimensional unitary space over $ F_{q^{2}} $. The inclusion graph of the totally isotropic subspace of $ \mathbb{U}_{n} $, written as $ \mathcal{I}n(\mathbb{U}_{n}) $, is a graph which has all totally isotropic subspaces of $ \mathbb{U}_{n} $ as its vertices and two distinct, totally isotropic subspaces $ U_{1} $, $ U_{2} $ of $ \mathbb{U}_{n} $ which are adjacent if and only if $ U_{1}\subset U_{2} $ or $ U_{2}\subset U_{1} $. In this paper, all automorphisms of $ \mathcal{I}n(\mathbb{U}_{n}) $ are determined definitely.

References

[1]	Z. Wan, Geometry of classical groups over finite fields, 2 Eds., Science Press, 2002.
[2]	A. Majidinya, Automorphisms of linear functional graphs over vector spaces, Linear Multilinear Algebra, 70 (2022), 5681–5697. https://doi.org/10.1080/03081087.2021.1923630 doi: 10.1080/03081087.2021.1923630
[3]	X. Wang, D. Wong, Automorphism group of the subspace inclusion graph of a vector space, Bull. Malays. Math. Sci. Soc., 42 (2019), 2213–2224. https://doi.org/10.1007/s40840-017-0597-2 doi: 10.1007/s40840-017-0597-2
[4]	X. Wang, D. Wong, D. Sun, Automorphisms and domination numbers of transformation graphs over vector spaces, Linear Multilinear Algebra, 67 (2019), 1350–1363. https://doi.org/10.1080/03081087.2018.1452890 doi: 10.1080/03081087.2018.1452890
[5]	Z. Tang, Z. Wan, Symplectic graphs and their automorphisms, Eur. J. Comb., 27 (2006), 38–50. https://doi.org/10.1016/j.ejc.2004.08.002 doi: 10.1016/j.ejc.2004.08.002
[6]	Z. Gu, Z. Wan, Orthogonal graphs of odd characteristic and their automorphisms, Finite Fields Appl., 14 (2008), 291–313. https://doi.org/10.1016/j.ffa.2006.12.001 doi: 10.1016/j.ffa.2006.12.001
[7]	Z. Wan, K. Zhou, Orthogonal graphs of characteristic $2$ and their automorphisms, Sci. China Ser. A Math., 52 (2009), 361–380. https://doi.org/10.1007/s11425-009-0018-6 doi: 10.1007/s11425-009-0018-6
[8]	S. Zhao, H. Zhang, J. Nan, G. Tang, Orthogonal inner product graphs over finite fields of odd characteristic, J. Math., 2023 (2023), 12. https://doi.org/10.1155/2023/6811540 doi: 10.1155/2023/6811540
[9]	Z. Gu, Z. Wan, K. Zhou, Automorphisms of subconstituents of unitary graphs over finite fields, Linear Multilinear Algebra, 64 (2016), 1833–1852. https://doi.org/10.1080/03081087.2015.1122721 doi: 10.1080/03081087.2015.1122721
[10]	S. Zhao, J. Nan, Unitary inner product graphs and their automorphisms, J. Algebra Appl., 22 (2023), 2350206. https://doi.org/10.1142/S0219498823502067 doi: 10.1142/S0219498823502067
[11]	Z. Wan, K. Zhou, Unitary graphs and their automorphisms, Ann. Comb., 14 (2010), 367–395. https://doi.org/10.1007/s00026-010-0065-2 doi: 10.1007/s00026-010-0065-2
[12]	J. He, G. Zhang, Automorphisms of symplectic totally isotropic subspace inclusion graph, J. Algebra Appl., 23 (2024), 2450024. https://doi.org/10.1142/S0219498824500245 doi: 10.1142/S0219498824500245
[13]	J. A. Body, U. S. R. Murty, Graph theory with applications, Elsevier Science Publishing Co., Inc, 1982.

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