Metric Bases of Barycentric and Matching Subdivisions of Zero-Divisor Graphs
Abstract
In this paper, we study metric bases and related metric properties for barycentric and partial matching subdivisions of the zero-divisor graph of ℤpq, where p and q are distinct odd primes with q>p. We first recall the natural partition of the zero-divisor graph into the two prime classes and then give a detailed characterization of those subsets of BS(Γ(ℤpq)) that form metric bases when q≥ 2p−1. The proof is expanded by separating the role of closed neighborhoods, rows of subdivision vertices, and forbidden twin configurations. We then investigate M-subdivision graphs obtained by subdividing selected edges of Γ(ℤpq). In addition to the lower bounds for subdivisions of p−3 and p−2 edges, we prove an exact formula for matching subdivisions of arbitrary size r, 0≤ r≤ p−2, namely dim(Gr)=p+q−r−4. Several consequences are included to illustrate how a small matching subdivision can reduce the localization cost of the original zero-divisor network.
Related
- Polytopal bases for barycentric subdivisions
- Zero-divisor graphs and zero-divisor real rank of C*-algebras
- Multiplicity-One Structure in Graded Zero-Divisor Graphs
- Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure
- On $4$-perfect-matching covers of cubic graphs with two adjacent odd circuits in a $2$-factor
BibTeX
@misc{vidya2026metric,
title = {{Metric Bases of Barycentric and Matching Subdivisions of Zero-Divisor Graphs}},
author = {Vidya S and Prasanna Poojary and Bilal Ahmad Rather and Vadiraja Bhatta G R},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09896},
eprint = {2609.09896},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09896},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Metric Bases of Barycentric and Matching Subdivisions of Zero-Divisor Graphs
AU - Vidya S
AU - Prasanna Poojary
AU - Bilal Ahmad Rather
AU - Vadiraja Bhatta G R
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09896
UR - https://arxiv.org/abs/2609.09896
AB - In this paper, we study metric bases and related metric properties for barycentric and partial matching subdivisions of the zero-divisor graph of ℤ_pq, where p and q are distinct odd primes with q>p. We first recall the natural partition of the zero-divisor graph into the two prime classes and then give a detailed characterization of those subsets of BS(Γ(ℤ_pq)) that form metric bases when q≥ 2p−1. The proof is expanded by separating the role of closed neighborhoods, rows of subdivision vertices, and forbidden twin configurations. We then investigate M-subdivision graphs obtained by subdividing selected edges of Γ(ℤ_pq). In addition to the lower bounds for subdivisions of p−3 and p−2 edges, we prove an exact formula for matching subdivisions of arbitrary size r, 0≤ r≤ p−2, namely dim(G_r)=p+q−r−4. Several consequences are included to illustrate how a small matching subdivision can reduce the localization cost of the original zero-divisor network.
ER -