arXiv subjects

Vidya S

Publications and source records attributed to Vidya S.

2 recordsLinked to original sources

arXiv2026

Metric Bases of Barycentric and Matching Subdivisions of Zero-Divisor Graphs

In this paper, we study metric bases and related metric properties for barycentric and partial matching subdivisions of the zero-divisor graph of $\mathbb Z_{pq}$, 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 $BS(\Gamma(\mathbb Z_{pq}))$ that form metric bases when $q\geq 2p-1$. 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 $\Gamma(\mathbb Z_{pq})$. 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 $r$, $0\leq r\leq p-2$, 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.

math.CO↗

arXiv2026

Fault-tolerant metric basis and dimension of barycentric subdivision of zero divisor graphs

The undirected zero divisor graph of a commutative ring with unity $ R $, denoted by $ \Gamma(R) = (V(\Gamma(R)), E(\Gamma(R))) $. The vertex set $ V(\Gamma(R)) $ consists of all the non-zero zero-divisors of $ R $. The edge set $ E(\Gamma(R)) $ is defined by the set $ \{ e = a_1 a_2 \mid a_1 \cdot a_2 = 0 \text{ and } a_1, a_2 \in V(\Gamma(R)) \} $. The barycentric subdivision of $\Gamma$ is the process of subdividing each edge by inserting new vertex in the graph $\Gamma$. In this article, we have focused on the fault-tolerant metric dimension of the barycentric subdivision of zero divisor graph of the group of integers modulo $ n $, represented by $ fdim(BS(\Gamma(\mathbb{Z}_n )$, where $ n = pq $; $ p $ and $ q $ are distinct odd primes with $ q > p $. We also demonstrate that $ fdim(BS(\Gamma(\mathbb{Z}_n) \geq q - 1 $ for every $ n = pq $, where $ p $ and $ q $ are any distinct odd primes with $ q > p $.

math.CO↗