Bounds on the Energy of Zero-divisor Graph of Quotient Ring and Its Topological Indices

In this paper, we study the zero-divisor graph of \( \mathbb{Z}_{\powerset}[x]/\langle x^5 \rangle \) for prime number \( \powerset \), denoted as \(\Gamma(\mathbb{Z}_{\powerset}[x]/\langle x^5 \rangle) \), including its energy and topological indices. Specifically, we provide bounds of the energy for \(\Gamma(\mathbb{Z}_{\powerset}[x]/\langle x^5 \rangle) \) and show that these bounds are numerically close to the actual energy value. Furthermore, we determine the topological indices of \(\Gamma(\mathbb{Z}_{\powerset}[x]/\langle x^5 \rangle) \), including the topological indices based on distance and degree. We also perform numerical simulations of the topological indices for various prime numbers \( \powerset \).