F1000Research · 2026
A Study on the Structure of Ideal-Based Non-Zero Divisor Graphs Associated with Z n.
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Kadem Sameer, Abd Aubad Ali
Аннотация
BACKGROUND: 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 ∅ I ( Z n ) , 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.
METHODS: Using combinatorial graph theory and modular arithmetic, we analyze fundamental properties of ∅ I ( Z n ) . Vertex degrees, connectivity, and cut-sets are characterized using divisibility conditions and the Euler totient function ϕ ( n ) . 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.
RESULTS: We establish necessary and sufficient conditions for the connectivity of ∅ I ( Z n ) , proving it is connected for all n ≥ 10 and any non-zero proper ideal I . For prime n ∉ { 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 = xy . Moreover, the domination number γ ( ∅ I ( Z n ) )=1 and girth gr ( ∅ I ( Z n ) )=3 is established for n ≥ 10 . General expressions for Zagreb and Randić indices are derived, directly linking graph invariants to n and d .
CONCLUSIONS: The graph ∅ I ( Z n ) 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.
METHODS: Using combinatorial graph theory and modular arithmetic, we analyze fundamental properties of ∅ I ( Z n ) . Vertex degrees, connectivity, and cut-sets are characterized using divisibility conditions and the Euler totient function ϕ ( n ) . 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.
RESULTS: We establish necessary and sufficient conditions for the connectivity of ∅ I ( Z n ) , proving it is connected for all n ≥ 10 and any non-zero proper ideal I . For prime n ∉ { 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 = xy . Moreover, the domination number γ ( ∅ I ( Z n ) )=1 and girth gr ( ∅ I ( Z n ) )=3 is established for n ≥ 10 . General expressions for Zagreb and Randić indices are derived, directly linking graph invariants to n and d .
CONCLUSIONS: The graph ∅ I ( Z n ) 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.