On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring Zp1T1p2T2

The zero-divisor graph of a commutative ring R with a nonzero identity, denoted by Γ (R), is an undirected graph where the vertex set Z (R) * consists of all nonzero zero-divisors of R. Two distinct vertices a and b in Γ (R) 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 Γ (Zn) for specific values of n in the set pq, p2q, p3, p4, where p and q are distinct primes satisfying p<q. Motivated by their work, this article investigates the normalized Laplacian spectrum of Γ (Zn) for a more general class of n, where n is represented as p1T1p2T2, with p1 and p2 being distinct primes (p1<p2), and T1, T2 are positive integers.