Exploring the zero-divisor graph over commutative ring: topological examine of algebraic structure

A commutative ring over a graph was first studied by Beck in 1988. The properties of zero-divisor graphs of commutative and non-commutative rings were then studied extensively. The commutative ring over graph is also widely used in robotics, communication theory, and elliptic curve cryptography. In this paper, we derive a few degree-based topological expressions namely, the first Zagreb index and the forgotten index of zero-divisor graphs of \({\mathbb {Z}}_p^n\) ${\mathbb{Z}}_p^n$ , \({\mathbb {Z}}_q\times {\mathbb {Z}}_p^n\) ${\mathbb{Z}}_q\times {\mathbb{Z}}_p^n$ and \({\mathbb {Z}}_{q^2}\times {\mathbb {Z}}_p^n\) ${\mathbb{Z}}_{q^2}\times {\mathbb{Z}}_p^n$ where p and q are primes. The numerical expression of the zero-divisor graphs is analysed and compared the indices graphically and numerically.