On properties and topological indices of the zero divisor graph for direct product of some commutative rings

Graph-theoretic representations of algebraic structures provide useful tools for studying algebraic properties and degree-distance-based invariants that arise, for example, in mathematical chemistry. Motivated by these applications and by recent work on zero divisor graphs over modular rings, this paper investigated the zero divisor graph $\mathrm{\Gamma }(R)$ of a commutative ring $R$ in the sense of Anderson and Livingston, where vertices are the zero divisors and two distinct vertices are adjacent when their product is zero. We focused on rings of the form ${\mathbb{Z}}_{p^m}\times {\mathbb{Z}}_{q^n}$, with $p,q$ primes and $m,n\in \mathbb{N}$. This research employed a literature review combined with a deductive method, drawing specific conclusions from well-established algebraic and graph-theoretic principles. By explicitly describing the zero divisors of $R$ and partitioning the vertex set according to greatest common divisor conditions, we determined several structural properties of $\mathrm{\Gamma }({\mathbb{Z}}_{p^m}\times {\mathbb{Z}}_{q^n})$. Furthermore, we obtained closed formulas for the general zeroth-order Randić index, the eccentric connectivity index, and the Schultz index of this graph. These results extended previous studies on zero divisor graphs over integers modulo rings and yield explicit families of graphs with exactly computable topological indices.