On the Steiner radial number of the zero-divisor graph for finite commutative rings ℤ _n

Abstract

Graphs serve as a bridge between algebra and numerous mathematical fields, including combinatorics, number theory, and computer science. Let Z(R) represent the set of zero-divisors in a commutative ring R. The zero-divisor graph of R, denoted by Γ(R), is an undirected graph with vertex set Z(R)^∗ = Z(R) \{0}, where two vertices are adjacent if their product is zero. The Steiner tree and Steiner radial number are fundamental concepts in network optimization, offering cost-efficient strategies for connecting multiple points with minimal total distance. In this study, we compute the m-eccentricity of all vertices and determine the Steiner radial number of the zero-divisor graph of ℤ _n . Additionally, for some positive integer m ≥ 2, we establish the existence of non-isomorphic zero-divisor graphs with Steiner radial number m. This work not only enhances the understanding of zero-divisor graphs but also provides a foundation for future studies in algebraic graph theory and network optimization.