Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 6 2 2025 08 01 Adjacency ‎s‎pectrum and energy of the exact zero-divisor graph of $\mathbb{Z}_n‏$ 149 158 233178 10.30504/jims.2025.546348.1284 EN S. K. Babariya Department of‎ ‎Mathematics, Dr‎. ‎Subhash University‎, ‎P.O‎. ‎Box 362001, Junagadh‎, ‎India. P. T. Lalchandani Department of‎ ‎Mathematics, Dr‎. ‎Subhash University‎, ‎P.O‎. ‎Box 362001, Junagadh‎, ‎India. Journal Article 2025 09 10 This paper investigates the exact zero-divisor graph $E\Gamma(R)$ of a commutative ring‎, ‎with particular focus on $R=\mathbb{Z}_n$‎. ‎We describe a partition of the vertex set of $E\Gamma(\mathbb{Z}_n)$ based on the greatest common divisors with $n$‎, ‎which provides structural information about its components‎. ‎The adjacency spectra and energies of $E\Gamma(\mathbb{Z}_{p^{2m+1}})$ and $E\Gamma(\mathbb{Z}_{p^{2m}})$ are computed explicitly‎, ‎and their asymptotic behaviors are compared‎. ‎The results reveal clear structural differences between the odd and even power cases‎. &lrm Exact zero-divisor graph&lrm adjacency matrix&lrm , &lrm Generalized Join Graph https://jims.ims.ir/article_233178_46a8603517fc211daa67fdfce0d94897.pdf