A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Z _n

The study of algebraic structures through graph-theoretic representations provides a powerful visual and combinatorial framework for analyzing ring-theoretic properties. The ideal-based non-zero divisor graph ${\varnothing }_I\left(Z_n\right)$ , constructed from the ring of integers modulo $n$ with respect to a proper ideal $I$ . This graph extends the classical zero-divisor graph framework and serves as a visual and structural invariant for analyzing ideal interactions in finite commutative rings.

Using combinatorial graph theory and modular arithmetic, we analyze fundamental properties of ${\varnothing }_I\left(Z_n\right)$ . Vertex degrees, connectivity, and cut-sets are characterized using divisibility conditions and the Euler totient function $\varphi \left(n\right)$ . The analysis distinguishes cases based on the parity and primality of $n$ , as well as the generator of $I$ . Topological indices, including the Zagreb and Randić indices, are formulated to quantify structural complexity.

We establish necessary and sufficient conditions for the connectivity of ${\varnothing }_I\left(Z_n\right)$ , proving it is connected for all $n\ge 10$ and any non-zero proper ideal $I$ . For prime $n\notin \{2,3\}$ , the graph is shown to be complete. General formulas are provided for calculating vertex degrees based on $\gcd (x,d)$ where $I=<d>$ . Furthermore, the structure and computation cut-sets are characterized for $Z_{p^2}$ and composite $n=\mathrm{xy}$ . Moreover, the domination number $\gamma$ ( ${\varnothing }_I\left(Z_n\right)$ )=1 and girth gr ( ${\varnothing }_I\left(Z_n\right)$ )=3 is established for $n\ge 10$ . General expressions for Zagreb and Randić indices are derived, directly linking graph invariants to $n$ and $d.$

The graph ${\varnothing }_I\left(Z_n\right)$ serves as an effective combinatorial invariant for studying the interplay between ideals and zero-divisor structure in $Z_n$ . These results establish systematic connections between ring-theoretic properties and graph parameters, enabling both qualitative and quantitative analysis through connectivity, degree distributions, cut-sets, and topological indices.