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Computing metric dimension of compressed zero divisor graphs associated to rings
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| Zeitschriftentitel: | Acta Universitatis Sapientiae, Mathematica |
|---|---|
| Personen und Körperschaften: | , |
| In: | Acta Universitatis Sapientiae, Mathematica, 10, 2018, 2, S. 298-318 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
Walter de Gruyter GmbH
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| Schlagwörter: |
| author_facet |
Pirzada, S. Bhat, M. Imran Pirzada, S. Bhat, M. Imran |
|---|---|
| author |
Pirzada, S. Bhat, M. Imran |
| spellingShingle |
Pirzada, S. Bhat, M. Imran Acta Universitatis Sapientiae, Mathematica Computing metric dimension of compressed zero divisor graphs associated to rings General Mathematics |
| author_sort |
pirzada, s. |
| spelling |
Pirzada, S. Bhat, M. Imran 2066-7752 Walter de Gruyter GmbH General Mathematics http://dx.doi.org/10.2478/ausm-2018-0023 <jats:title>Abstract</jats:title> <jats:p>For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph Γ<jats:sub>E</jats:sub>(R) with vertex set Z(R<jats:sub>E</jats:sub>) \ {[0]} = R<jats:sub>E</jats:sub> \ {[0], [1]} defined by R<jats:sub>E</jats:sub> = {[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 Γ<jats:sub>E</jats:sub>(R), the relationship of metric dimension between Γ<jats:sub>E</jats:sub>(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of Γ<jats:sub>E</jats:sub>(R). We provide a formula for the number of vertices of the family of graphs given by Γ<jats:sub>E</jats:sub>(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of Γ<jats:sub>E</jats:sub>(R).</jats:p> Computing metric dimension of compressed zero divisor graphs associated to rings Acta Universitatis Sapientiae, Mathematica |
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Acta Universitatis Sapientiae, Mathematica |
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| title |
Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_unstemmed |
Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_full |
Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_fullStr |
Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_full_unstemmed |
Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_short |
Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_sort |
computing metric dimension of compressed zero divisor graphs associated to rings |
| topic |
General Mathematics |
| url |
http://dx.doi.org/10.2478/ausm-2018-0023 |
| publishDate |
2018 |
| physical |
298-318 |
| description |
<jats:title>Abstract</jats:title>
<jats:p>For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph Γ<jats:sub>E</jats:sub>(R) with vertex set Z(R<jats:sub>E</jats:sub>) \ {[0]} = R<jats:sub>E</jats:sub> \ {[0], [1]} defined by R<jats:sub>E</jats:sub> = {[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 Γ<jats:sub>E</jats:sub>(R), the relationship of metric dimension between Γ<jats:sub>E</jats:sub>(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of Γ<jats:sub>E</jats:sub>(R). We provide a formula for the number of vertices of the family of graphs given by Γ<jats:sub>E</jats:sub>(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of Γ<jats:sub>E</jats:sub>(R).</jats:p> |
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| description | <jats:title>Abstract</jats:title> <jats:p>For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph Γ<jats:sub>E</jats:sub>(R) with vertex set Z(R<jats:sub>E</jats:sub>) \ {[0]} = R<jats:sub>E</jats:sub> \ {[0], [1]} defined by R<jats:sub>E</jats:sub> = {[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 Γ<jats:sub>E</jats:sub>(R), the relationship of metric dimension between Γ<jats:sub>E</jats:sub>(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of Γ<jats:sub>E</jats:sub>(R). We provide a formula for the number of vertices of the family of graphs given by Γ<jats:sub>E</jats:sub>(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of Γ<jats:sub>E</jats:sub>(R).</jats:p> |
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| spelling | Pirzada, S. Bhat, M. Imran 2066-7752 Walter de Gruyter GmbH General Mathematics http://dx.doi.org/10.2478/ausm-2018-0023 <jats:title>Abstract</jats:title> <jats:p>For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph Γ<jats:sub>E</jats:sub>(R) with vertex set Z(R<jats:sub>E</jats:sub>) \ {[0]} = R<jats:sub>E</jats:sub> \ {[0], [1]} defined by R<jats:sub>E</jats:sub> = {[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 Γ<jats:sub>E</jats:sub>(R), the relationship of metric dimension between Γ<jats:sub>E</jats:sub>(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of Γ<jats:sub>E</jats:sub>(R). We provide a formula for the number of vertices of the family of graphs given by Γ<jats:sub>E</jats:sub>(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of Γ<jats:sub>E</jats:sub>(R).</jats:p> Computing metric dimension of compressed zero divisor graphs associated to rings Acta Universitatis Sapientiae, Mathematica |
| spellingShingle | Pirzada, S., Bhat, M. Imran, Acta Universitatis Sapientiae, Mathematica, Computing metric dimension of compressed zero divisor graphs associated to rings, General Mathematics |
| title | Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_full | Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_fullStr | Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_full_unstemmed | Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_short | Computing metric dimension of compressed zero divisor graphs associated to rings |
| title_sort | computing metric dimension of compressed zero divisor graphs associated to rings |
| title_unstemmed | Computing metric dimension of compressed zero divisor graphs associated to rings |
| topic | General Mathematics |
| url | http://dx.doi.org/10.2478/ausm-2018-0023 |