The Steiner antipodal number of zero-divisor graphs of finite commutative rings
Abstract
Let $ R $ be a finite commutative ring with unity. The zero-divisor graph $ \Gamma(R) $ is defined such that its vertex set comprises all nonzero zero-divisors of $ R $, with two distinct vertices being adjacent if and only if their product is zero. This study provides a closed-form expression for the $ n $-eccentricity of each vertex and computes the Steiner antipodal number of $ \Gamma(R) $ under the following conditions: (ⅰ) $ R = \mathbb{Z}_m $, (ⅱ) $ R $ is a reduced ring, and (ⅲ) $ R $ is a finite direct product of rings of the form $ \mathbb{Z}_m $. Moreover, we establish the existence of a zero-divisor graph with a Steiner antipodal number equal to some positive integer $ m $.
Fields of Study & Subjects
Rings
Modules
and Algebras
Algebraic Geometry and Number Theory
Finite Group Theory Research
Publication details
- Journal / source
- AIMS Mathematics
- Publisher
- American Institute of Mathematical Sciences (AIMS)
- Volume
- 11
- Issue
- 2
- Pages
- 3512–3533
- Publication date
- 1 January 2026