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

We introduce the nilpotent graph ΓN(L) of a finite-dimensional Lie superalgebra L = L0̄L1̄ 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, Fq), we show that ΓN(L) is the disjoint union of q+1 complete graphs Kq(q−1), each having q(q−1) vertices. We characterize the bipartiteness of ΓN(L), demonstrating that it is bipartite if and only if the odd component L1̄ ⊆ N (L). We also analyze connectivity, diameter, clique number, and chromatic number for Lie superalgebras such as sl(1|1, Fq) and investigate direct sums and complement graphs. SageMath algorithms are provided to compute ΓN(L) and its invariants, linking the algebraic structure of Lie superalgebras to graph theory and emphasizing the role of Z2-grading in topological properties. Open problems on higher-dimensional structures and spectral properties are proposed.