Computing metric dimension of compressed zero divisor graphs associated to rings

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Zeitschriftentitel:	Acta Universitatis Sapientiae, Mathematica
Personen und Körperschaften:	Pirzada, S., Bhat, M. Imran
In:	Acta Universitatis Sapientiae, Mathematica, 10, 2018, 2, S. 298-318
Format:	E-Article
Sprache:	Englisch
veröffentlicht:	Walter de Gruyter GmbH
Schlagwörter:	General Mathematics

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Zusammenfassung:	<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>
Umfang:	298-318
ISSN:	2066-7752
DOI:	10.2478/ausm-2018-0023