@article{Hummdi2025, 
author = {Ali Yahya Hummdi and Kholood Alnefaie and Giovanni Scudo and Mohammad Shane Alam},
title = {Nilpotent graphs of Lie superalgebras: structure and graph-theoretic properties},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {12},
pages = {28451-28469},
keywords = {Lie superalgebra, nilpotent graph, nilpotentizer, graph invariants, triangular matrices},
url = {https://www.sciopen.com/article/10.3934/math.20251252},
doi = {10.3934/math.20251252},
abstract = {We introduce the nilpotent graph        Γ    N    (  L  ) of a finite-dimensional Lie superalgebra    L  =      L                            0          ¯                      ⊕      L                            1          ¯                     over a field        F  , with vertices consisting of non-nilpotent elements and edges connecting pairs that generate nilpotent subsuperalgebras. We prove that the nilpotentizer        N    (  L  ) coincides with the hypercenter        Z    ∗    (  L  ) when    char  (      F    )  =  0. For the triangular Lie superalgebra        t    (  2  ,            F        q    ), we show that        Γ          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        Γ    N    (  L  ), demonstrating that it is bipartite if and only if the odd component        L                            1          ¯                      ⊆      N    (  L  ). We also analyze connectivity, diameter, clique number, and chromatic number for Lie superalgebras such as        s    l    (  1      |    1  ,            F        q    ) and investigate direct sums and complement graphs. SageMath algorithms are provided to compute        Γ    N    (  L  ) and its invariants, linking the algebraic structure of Lie superalgebras to graph theory and emphasizing the role of              Z        2  -grading in topological properties. Open problems on higher-dimensional structures and spectral properties are proposed.}
}