Nilpotent graphs of Lie superalgebras: structure and graph-theoretic properties

Ali Yahya Hummdi; Kholood Alnefaie; Giovanni Scudo; Mohammad Shane Alam; Ali Yahya Hummdi; Kholood Alnefaie; Giovanni Scudo; Mohammad Shane Alam

doi:10.3934/math.20251252

AIMS Mathematics

2025, Volume 10, Issue 12: 28451-28469. doi: 10.3934/math.20251252

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Research article

Nilpotent graphs of Lie superalgebras: structure and graph-theoretic properties

1.
Department of Mathematics, College of Science, King Khalid University, Abha 61471, Saudi Arabia
2.
Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia
3.
Department of Engineering, University of Messina, 98166 Messina, Italy
4.
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received: 21 July 2025 Revised: 20 November 2025 Accepted: 25 November 2025 Published: 03 December 2025
MSC : 17B05, 05C25, 17B20, 68W30

We introduce the nilpotent graph $ \Gamma_N(L) $ of a finite-dimensional Lie superalgebra $ L = L_{\bar{0}} \oplus L_{\bar{1}} $ over a field $ \mathbb{F} $, with vertices consisting of non-nilpotent elements and edges connecting pairs that generate nilpotent subsuperalgebras. We prove that the nilpotentizer $ \mathscr{N}(L) $ coincides with the hypercenter $ Z^*(L) $ when $ \text{char}(\mathbb{F}) = 0 $. For the triangular Lie superalgebra $ \mathfrak{t}(2, \mathbb{F}_q) $, we show that $ \Gamma_{N}(L) $ is the disjoint union of $ q+1 $ complete graphs $ K_{q(q-1)} $, each having $ q(q-1) $ vertices. We characterize the bipartiteness of $ \Gamma_N(L) $, demonstrating that it is bipartite if and only if the odd component $ L_{\bar{1}} \subseteq \mathscr{N}(L) $. We also analyze connectivity, diameter, clique number, and chromatic number for Lie superalgebras such as $ \mathfrak{sl}(1|1, \mathbb{F}_q) $ and investigate direct sums and complement graphs. SageMath algorithms are provided to compute $ \Gamma_N(L) $ and its invariants, linking the algebraic structure of Lie superalgebras to graph theory and emphasizing the role of $ \mathbb{Z}_2 $-grading in topological properties. Open problems on higher-dimensional structures and spectral properties are proposed.
- Lie superalgebra,
- nilpotent graph,
- nilpotentizer,
- graph invariants,
- triangular matrices
Citation: Ali Yahya Hummdi, Kholood Alnefaie, Giovanni Scudo, Mohammad Shane Alam. Nilpotent graphs of Lie superalgebras: structure and graph-theoretic properties[J]. AIMS Mathematics, 2025, 10(12): 28451-28469. doi: 10.3934/math.20251252

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Abstract

We introduce the nilpotent graph $ \Gamma_N(L) $ of a finite-dimensional Lie superalgebra $ L = L_{\bar{0}} \oplus L_{\bar{1}} $ over a field $ \mathbb{F} $, with vertices consisting of non-nilpotent elements and edges connecting pairs that generate nilpotent subsuperalgebras. We prove that the nilpotentizer $ \mathscr{N}(L) $ coincides with the hypercenter $ Z^*(L) $ when $ \text{char}(\mathbb{F}) = 0 $. For the triangular Lie superalgebra $ \mathfrak{t}(2, \mathbb{F}_q) $, we show that $ \Gamma_{N}(L) $ is the disjoint union of $ q+1 $ complete graphs $ K_{q(q-1)} $, each having $ q(q-1) $ vertices. We characterize the bipartiteness of $ \Gamma_N(L) $, demonstrating that it is bipartite if and only if the odd component $ L_{\bar{1}} \subseteq \mathscr{N}(L) $. We also analyze connectivity, diameter, clique number, and chromatic number for Lie superalgebras such as $ \mathfrak{sl}(1|1, \mathbb{F}_q) $ and investigate direct sums and complement graphs. SageMath algorithms are provided to compute $ \Gamma_N(L) $ and its invariants, linking the algebraic structure of Lie superalgebras to graph theory and emphasizing the role of $ \mathbb{Z}_2 $-grading in topological properties. Open problems on higher-dimensional structures and spectral properties are proposed.

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