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

We introduce the nilpotent graph ${\mathrm{\Gamma }}_N(L)$ of a finite-dimensional Lie superalgebra $L=L_{\overline{0}}\oplus L_{\overline{1}}$ 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 $\mathcal{N}(L)$ coincides with the hypercenter $Z^{\ast }(L)$ when $\text{char}(\mathbb{F})=0$. For the triangular Lie superalgebra $\mathfrak{t}(2,{\mathbb{F}}_q)$, we show that ${\mathrm{\Gamma }}_N(L)$ is the disjoint union of $q+1$ complete graphs $K_{q(q-1)}$, each having $q(q-1)$ vertices. We characterize the bipartiteness of ${\mathrm{\Gamma }}_N(L)$, demonstrating that it is bipartite if and only if the odd component $L_{\overline{1}}\subseteq \mathcal{N}(L)$. We also analyze connectivity, diameter, clique number, and chromatic number for Lie superalgebras such as $\mathfrak{s}\mathfrak{l}(1|1,{\mathbb{F}}_q)$ and investigate direct sums and complement graphs. SageMath algorithms are provided to compute ${\mathrm{\Gamma }}_N(L)$ 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.