Sombor index and eigenvalues of weakly zero-divisor graph of commutative rings

The weakly zero-divisor graph WΓ(R) of a commutative ring R is the simple undirected graph whose vertices are nonzero zero-divisors of R and two distinct vertices x, y are adjacent if and only if there exist w∈ ann(x) and z∈ ann(y) such that $wz =0$. In this paper, we determine the Sombor index for the weakly zero-divisor graph of the integers modulo ring Zₙ. Furthermore, we investigate the Sombor spectrum and establish bounds for the Sombor energy of the weakly zero-divisor graph of Zₙ.