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

Graphs serve as a bridge between algebra and numerous mathematicalfields, including combinatorics, number theory, and computer science. Let Z(R) represent the set of zero-divisors in a commutative ring R. The zerodivisor 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 Zn. 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 zerodivisor graphs but also provides a foundation for future studies in algebraic graph theory and network optimization.