Malaysian Journal of Mathematical Sciences, March 2026, Vol. 20, No. 1
On the Spectrum of Generalized Zero-divisor Graph of the Ring $\mathbb Z_{p^{k_1}q^{k_2}r^{k_3}}$
Khairnar, A. and Lande, A.
Corresponding Email: anita7783@gmail.com
Received date: 6 January 2025
Accepted date: 28 July 2025
Abstract:
The generalized zero-divisor graph of a commutative ring $R$ denoted by $\Gamma'(R)$, is a simple (undirected) graph with the vertex set consisting of all nonzero zero-divisors in $R$ and two vertices $x$ and $y$ are adjacent if $x^ny=0$ or $y^nx=0$ for some positive integer $n$. For the distinct primes $p,q,r$ and positive integers $k_1,k_2,k_3$, we determine the adjacency matrix and eigenvalues of $\Gamma' \left(\mathbb Z_{p^{k_1}q^{k_2}r^{k_3}} \right)$. Also, we calculate the clique number, diameter, girth and stability number of $\Gamma' \left(\mathbb Z_{p^{k_1}q^{k_2}r^{k_3}} \right)$ and verify Beck's conjecture for $\Gamma' \left(\mathbb Z_{p^{k_1}q^{k_2}r^{k_3}} \right)$.
Keywords: generalized zero-divisor graph; eigenvalues; adjacency matrix.
