On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring <semantics><mstyle mathvariant="bold">Z</mstyle><msubsup>p1T1</msubsup><msubsup>p2T2</msubsup></semantics> - Ali Al Khabyah; Nazim; Nadeem Ur Rehman (Axioms)
The zero-divisor graph of a commutative ring <semantics>R</semantics> with a nonzero identity, denoted by <semantics>Γ(R)</semantics>, is an undirected graph where the vertex set <semantics>Z(R)*</semantics> consists of all nonzero zero-divisors of <semantics>R</semantics>. Two distinct vertices a and b in <semantics>Γ(R)</semantics> are adjacent if and only if <semantics>ab=0</semantics>. 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 <semantics>Γ(Zn)</semantics> for specific values of <semantics>n</semantics> in the set <semantics>{pq,p2q,p3,p4}</semantics>, where p and q are distinct primes satisfying <semantics>p<q</semantics>. Motivated by their work, this article investigates the normalized Laplacian spectrum of <semantics>Γ(Zn)</semantics> for a more general class of <semantics>n</semantics>, where <semantics>n</semantics> is represented as <semantics><msubsup>p1T1</msubsup><msubsup>p2T2</msubsup></semantics>, with <semantics>p1</semantics> and <semantics>p2</semantics> being distinct primes <semantics>(p1<p2)</semantics>, and <semantics>T1,T2</semantics> are positive integers.