Author

# M. Imran Bhat

Bio: M. Imran Bhat is an academic researcher from University of Kashmir. The author has contributed to research in topics: Metric dimension & Zero divisor. The author has an hindex of 3, co-authored 3 publications receiving 23 citations.

##### Papers

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01 Jun 2020TL;DR: For a commutative ring R with non-zero zero-divisor set (Z^*(R)) and vertex set Z = 0, where two distinct vertices x and y are adjacent if and only if x = 0 and y = 0 respectively, the clique number, degree of the vertices, size, metric dimension, upper dimension, automorphism group, Wiener index of the associated zero-Divisor graph of as mentioned in this paper.

Abstract: For a commutative ring R with non-zero zero-divisor set $$Z^*(R)$$, the zero-divisor graph of R is $$\varGamma (R)$$ with vertex set $$Z^*(R)$$, where two distinct vertices x and y are adjacent if and only if $$xy=0$$. The zero-divisor graph structure of $${\mathbb {Z}}_{p^n}$$ is described. We determine the clique number, degree of the vertices, size, metric dimension, upper dimension, automorphism group, Wiener index of the associated zero-divisor graph of $${\mathbb {Z}}_{p^n}$$. Further, we provide a partition of the vertex set of $$\varGamma ({\mathbb {Z}}_{p^n})$$ into distance similar equivalence classes and we show that in this graph the upper dimension equals the metric dimension. Also, we discuss similar properties of the compressed zero-divisor graph.

20 citations

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TL;DR: In this paper, the authors studied the metric dimension of the compressed zero divisor graph ΓE(R), the relationship of metric dimension between ΓR and Γ(R) and classified the rings with same or different metric dimension.

Abstract: Abstract For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph ΓE(R) with vertex set Z(RE) \\ {[0]} = RE \\ {[0], [1]} defined by RE = {[x] : x ∈ R}, where [x] = {y ∈ R : ann(x) = ann(y)} and the two distinct vertices [x] and [y] of Z(RE) are adjacent if and only if [x][y] = [xy] = [0], that is, if and only if xy = 0. In this paper, we study the metric dimension of the compressed zero divisor graph ΓE(R), the relationship of metric dimension between ΓE(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of ΓE(R). We provide a formula for the number of vertices of the family of graphs given by ΓE(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of ΓE(R).

14 citations

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TL;DR: In this paper, the strong metric dimension of zero-divisor graphs is studied by transforming the problem of finding the vertex cover number of a strong resolving graph into a more well-known problem.

Abstract: In this paper, we study the strong metric dimension of zero-divisor graph $\Gamma(R)$ associated to a ring $R$. This is done by transforming the problem into a more well-known problem of finding the vertex cover number $\alpha(G)$ of a strong resolving graph $G_{sr}$. We find the strong metric dimension of zero-divisor graphs of the ring $\mathbb{Z}_n$ of integers modulo $n$ and the ring of Gaussian integers $\mathbb{Z}_n[i]$ modulo $n$. We obtain the bounds for strong metric dimension of zero-divisor graphs and we also discuss the strong metric dimension of the Cartesian product of graphs.

9 citations

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01 Jun 2020TL;DR: For a commutative ring R with non-zero zero-divisor set (Z^*(R)) and vertex set Z = 0, where two distinct vertices x and y are adjacent if and only if x = 0 and y = 0 respectively, the clique number, degree of the vertices, size, metric dimension, upper dimension, automorphism group, Wiener index of the associated zero-Divisor graph of as mentioned in this paper.

Abstract: For a commutative ring R with non-zero zero-divisor set $$Z^*(R)$$, the zero-divisor graph of R is $$\varGamma (R)$$ with vertex set $$Z^*(R)$$, where two distinct vertices x and y are adjacent if and only if $$xy=0$$. The zero-divisor graph structure of $${\mathbb {Z}}_{p^n}$$ is described. We determine the clique number, degree of the vertices, size, metric dimension, upper dimension, automorphism group, Wiener index of the associated zero-divisor graph of $${\mathbb {Z}}_{p^n}$$. Further, we provide a partition of the vertex set of $$\varGamma ({\mathbb {Z}}_{p^n})$$ into distance similar equivalence classes and we show that in this graph the upper dimension equals the metric dimension. Also, we discuss similar properties of the compressed zero-divisor graph.

20 citations

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26 Feb 2021

TL;DR: In this paper, the structure and Laplacian spectrum of the zero-divisor graph Γ(Zn) for n =pN1qN2, where p

Abstract: Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖{0} be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p

19 citations

09 Apr 2020

TL;DR: In this article, a comprehensive survey on zero-divisor graphs of finite commutative rings is given, and the results on structural properties of these graphs are investigated, e.g.

Abstract: This article gives a comprehensive survey on zero-divisor graphs of finite commutative rings. We investigate the results on structural properties of these graphs.

13 citations