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

The graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mo>∅</mml:mo> <mml:mi>I</mml:mi></mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:msub><mml:mi>Z</mml:mi> <mml:mi>n</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:math> serves as an effective combinatorial invariant for studying the interplay between ideals and zero-divisor structure in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mspace></mml:mspace> <mml:msub><mml:mi>Z</mml:mi> <mml:mi>n</mml:mi></mml:msub> </mml:math> . 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.