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

TL;DR

This paper studies the structure of a specific graph derived from the ring of integers modulo n, using graph theory to analyze algebraic properties.

Contribution

The paper introduces and analyzes the ideal-based non-zero divisor graph ∅I(Zn), deriving new properties and formulas for its structure.

Findings

The graph ∅I(Zn) is connected for all n ≥ 10 and any non-zero proper ideal I.

For prime n ∉ {2,3}, the graph is complete.

General formulas for vertex degrees and topological indices are derived based on n and d.

Abstract

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(Zn)
, 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

∅I(Zn)
. 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.

We establish necessary and sufficient conditions for the connectivity of

∅I(Zn)
, 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

Zp2
and composite

n=xy
. Moreover, the domination number

γ
(

∅I(Zn)
)=1 and girth gr (

∅I(Zn)
)=3 is established for

n≥10
. General expressions for Zagreb and Randić indices are derived, directly linking graph invariants to

n
and

d.

The graph

∅I(Zn)
serves as an effective combinatorial invariant for studying the interplay between ideals and zero-divisor structure in

Zn
. 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.

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