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Vadiraja Bhatta G R

Publications and source records attributed to Vadiraja Bhatta G R.

5 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 $BS(Γ(\mathbb Z_{pq}))$ 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 $Γ(\mathbb Z_{pq})$. 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 \( Γ(R) = (V(Γ(R)), E(Γ(R))) \). The vertex set \( V(Γ(R)) \) consists of all the non-zero zero-divisors of \( R \). The edge set \( E(Γ(R)) \) is defined by the set \( \{ e = a_1 a_2 \mid a_1 \cdot a_2 = 0 \text{ and } a_1, a_2 \in V(Γ(R)) \} \). The barycentric subdivision of $Γ$ is the process of subdividing each edge by inserting new vertex in the graph $Γ$. 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 \( fdim(BS(Γ(\mathbb{Z}_n )\), where \( n = pq \); \( p \) and \( q \) are distinct odd primes with \( q > p \). We also demonstrate that \( fdim(BS(Γ(\mathbb{Z}_n) \geq q - 1 \) for every \( n = pq \), where \( p \) and \( q \) are any distinct odd primes with \( q > p \).

math.CO↗
arXiv2025

Lee Distance of cyclic codes of length $2^ς$ over $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$

Let $p$ be a prime number and $ς$ and $m$ be a positive integers. Let $\mathcal{R} = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$ ($u^3 = 0$). Cyclic codes of length $2^ς$ over $\mathcal{R}$ are precisely the ideals of the local ring $\frac{\mathcal{R}[x]}{\langle x^{2^ς}-1 \rangle}$. The Gray map from a code of Lee weight over $\mathbb{Z}_4$ to a code with Hamming weight over $\mathbb{F}_2$ is known to preserve weight. In this paper, we determine the Lee distance of cyclic codes of length $2^ς$ over $\mathcal{R}$.

math.CO↗
arXiv2024

Repeated-Root Constacyclic Codes of Length $3p^s$ over the Finite Non-Chain Ring $\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}$ and their Duals

This study aims to determine the algebraic structures of $α$-constacyclic codes of length $3p^s$ over the finite commutative non-chain ring $\mathcal{R}=\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}$, for a prime $p \neq 3.$ For the unit $α$, we consider two different instances: when $α$ is a cube in $\mathcal{R}$ and when it is not. Analyzing the first scenario is relatively easy. When $α$ is not a unit in $\mathcal{R}$, we consider several subcases and determine the algebraic structures of constacyclic codes in those cases. Further, we also provide the number of codewords and the duals of $α$-constacyclic codes.

math.CO↗