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

The weakly zero-divisor graph $W\mathrm{\Gamma }(R)$ 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 $w\in \mathrm{a}\mathrm{n}\mathrm{n}(x)$ and $z\in \mathrm{a}\mathrm{n}\mathrm{n}(y)$ such that $wz=0$. In this paper, we determine the Sombor index for the weakly zero-divisor graph of the integers modulo ring ${\mathbb{Z}}_n$. Furthermore, we investigate the Sombor spectrum and establish bounds for the Sombor energy of the weakly zero-divisor graph of ${\mathbb{Z}}_n$.