For a finite commutative ring $\mathfrak{R}$ with identity $1\ne 0$, the weakly zero-divisor graph of $\mathfrak{R}$ denoted as $W\Gamma \left(\mathfrak{R}\right)$ is a simple undirected graph having vertex set as a set of non-zero zero-divisors of $\mathfrak{R}$ and two distinct vertices $\mathfrak{a}$ and $\mathfrak{b}$ are adjacent if and only if there exist elements $r\in \mathrm{ann}\left(\mathfrak{a}\right)$ and $s\in \mathrm{ann}\left(\mathfrak{b}\right)$ satisfying the condition $rs=0$. The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph $W\Gamma \left(\mathfrak{R}\right)$. Specifically, the investigation is carried out on the weakly zero-divisor graph $W\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for various values of $\mathfrak{n}$.

The zero-divisor graph of a commutative ring $\mathfrak{R}$ with a nonzero identity, denoted by $\Gamma \left(\mathfrak{R}\right)$, is an undirected graph where the vertex set $Z{\left(\mathfrak{R}\right)}^{*}$ consists of all nonzero zero-divisors of $\mathfrak{R}$. Two distinct vertices a and b in $\Gamma \left(\mathfrak{R}\right)$ are adjacent if and only if $ab=0$. The normalized Laplacian spectrum of zero-divisor graphs has been studied extensively due to its algebraic and combinatorial significance. Notably, Pirzada and his co-authors computed the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for specific values of $\mathfrak{n}$ in the set $\{pq,p^{2}q,p^3,p^4\}$, where p and q are distinct primes satisfying $p<q$. Motivated by their work, this article investigates the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for a more general class of $\mathfrak{n}$, where $\mathfrak{n}$ is represented as $p_1^{T_1}{p}_2^{T_2}$, with $p_1$ and $p_2$ being distinct primes $(p_1<p_2)$, and $T_1,T_2$ are positive integers.

This paper extends the notions of n-derivations and n-automorphisms from Lie algebras to nest algebras via exponential mappings. We establish necessary and sufficient conditions for triangularity, and examine the preservation of the radical, center, and ideals under these higher-order algebraic transformations. The induced group structures of n-automorphisms are explicitly characterized, including inner and non-abelian components. Several concrete examples demonstrate the applicability and depth of the theoretical findings.