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John D. Lagrange - One of the best experts on this subject based on the ideXlab platform.
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some remarks on the compressed Zero Divisor graph
Journal of Algebra, 2016Co-Authors: David F. Anderson, John D. LagrangeAbstract:Abstract Let R be a commutative ring with 1 ≠ 0 . The Zero-Divisor graph Γ ( R ) of R is the (undirected) graph with vertices the nonZero Zero-Divisors of R, and distinct vertices r and s are adjacent if and only if r s = 0 . The relation on R given by r ∼ s if and only if ann R ( r ) = ann R ( s ) is an equivalence relation. The compressed Zero-Divisor graph Γ E ( R ) of R is the (undirected) graph with vertices the equivalence classes induced by ∼ other than [0] and [1], and distinct vertices [ r ] and [ s ] are adjacent if and only if r s = 0 . Let R E be the set of equivalence classes for ∼ on R. Then R E is a commutative monoid with multiplication [ r ] [ s ] = [ r s ] . In this paper, we continue our study of the monoid R E and the compressed Zero-Divisor graph Γ E ( R ) . We consider several equivalence relations on R and their corresponding graph-theoretic translations to Γ ( R ) . We also show that the girth of Γ E ( R ) is three if it contains a cycle and determine the structure of Γ E ( R ) when it is acyclic and the monoids R E when Γ E ( R ) is a star graph.
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characterizations of three classes of Zero Divisor graphs
Canadian Mathematical Bulletin, 2012Co-Authors: John D. LagrangeAbstract:The Zero-Divisor graph Γ(R) of a commutative ring R is the graph whose vertices consist of the nonZero Zero-Divisors of R such that distinct vertices x and y are adjacent if and only if xy = 0. In this paper, a characterization is provided for Zero-Divisor graphs of Boolean rings. Also, commutative rings R such that Γ(R) is isomorphic to the Zero-Divisor graph of a direct product of integral domains are classified, as well as those whose Zero-Divisor graphs are central vertex complete. School of Natural Sciences, Indiana University Southeast, New Albany, Indiana 47150, USA e-mail: lagrangej@lindsey.edu Received by the editors February 2, 2009; revised May 22, 2009. Published electronically May 30, 2011. AMS subject classification: 13A99, 13M99. 1
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On Realizing Zero-Divisor Graphs
Communications in Algebra, 2008Co-Authors: John D. LagrangeAbstract:An algorithm is presented for constructing the Zero-Divisor graph of a direct product of integral domains. Moreover, graphs which are realizable as Zero-Divisor graphs of direct products of integral domains are classified, as well as those of Boolean rings. In particular, graphs which are realizable as Zero-Divisor graphs of finite reduced commutative rings are classified.
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Complemented Zero-Divisor graphs and Boolean rings
Journal of Algebra, 2007Co-Authors: John D. LagrangeAbstract:Abstract For a commutative ring R, the Zero-Divisor graph of R is the graph whose vertices are the nonZero Zero-Divisors of R such that the vertices x and y are adjacent if and only if x y = 0 . In this paper, we classify the Zero-Divisor graphs of Boolean rings, as well as those of Boolean rings that are rationally complete. We also provide a complete list of those rings whose Zero-Divisor graphs have the property that every vertex is either an end or adjacent to an end.
Vinayak Joshi - One of the best experts on this subject based on the ideXlab platform.
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Zero-Divisor graphs and total coloring conjecture
Soft Computing, 2020Co-Authors: Nilesh Khandekar, Vinayak JoshiAbstract:In this paper, we prove that the Zero-Divisor graphs of a special class of pseudocomplemented posets satisfy the total coloring conjecture. Also, we determine the edge chromatic number of the Zero-Divisor graphs of this special class of pseudocomplemented posets. These results are applied to Zero-Divisor graphs of finite reduced commutative rings.
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Zero-Divisor graphs of lower dismantlable lattices II
Mathematica Slovaca, 2018Co-Authors: Avinash Patil, B. N. Waphare, Vinayak JoshiAbstract:Abstract In this paper, we continue our study of the Zero-Divisor graphs of lower dismantlable lattices that was started in [PATIL, A.—WAPHARE, B. N.—JOSHI, V.—POURALI, H. Y.: Zero-Divisor graphs of lower dismantlable lattices I, Math. Slovaca 67 (2017), 285–296]. The present paper mainly deals with an Isomorphism Problem for the Zero-Divisor graphs of lattices. In fact, we prove that the Zero-Divisor graphs of lower dismantlable lattices with the greatest element 1 as join-reducible are isomorphic if and only if the lattices are isomorphic.
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perfect Zero Divisor graphs
Discrete Mathematics, 2017Co-Authors: Avinash Patil, B. N. Waphare, Vinayak JoshiAbstract:In this article, we characterize various algebraic and order structures whose Zero-Divisor graphs are perfect graphs. We strengthen the result of Chenź(2003, Theorem 2.5) by providing a simpler proof of Beck's conjecture for the class of finite reduced rings (not necessarily commutative).
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the Zero Divisor graphs of boolean posets
Mathematica Slovaca, 2014Co-Authors: Vinayak Joshi, Anagha KhisteAbstract:In this paper, it is proved that if B is a Boolean poset and S is a bounded pseudocomplemented poset such that S\Z(S) = {1}, then Γ(B) ≌ Γ(S) if and only if B ≌ S. Further, we characterize the graphs which can be realized as Zero Divisor graphs of Boolean posets.
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On generalized Zero Divisor graph of a poset
Discrete Applied Mathematics, 2013Co-Authors: Vinayak Joshi, B. N. Waphare, H. Y. PouraliAbstract:In this paper, we introduce the generalized ideal based Zero Divisor graph of a poset P, denoted by G"I(P)@^. A representation theorem is obtained for generalized Zero Divisor graphs. It is proved that a graph is complete r-partite with r>=2 if and only if it is a generalized Zero Divisor graph of a poset. As a consequence of this result, we prove a form of a Beck's Conjecture for generalized Zero Divisor graphs of a poset. Further, it is proved that a generalized Zero Divisor graph G"{"0"}(P)@^ of a section semi-complemented poset P with respect to the ideal (0] is a complete graph.
David F. Anderson - One of the best experts on this subject based on the ideXlab platform.
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GENERALIZATIONS OF THE Zero-Divisor GRAPH
International Electronic Journal of Algebra, 2020Co-Authors: David F. Anderson, Grace McclurkinAbstract:Let $R$ be a commutative ring with $1 \neq 0$ and $Z(R)$ its set of Zero-Divisors. The Zero-Divisor graph of $R$ is the (simple) graph $\Gamma(R)$ with vertices $Z(R) \setminus \{0\}$, and distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. In this paper, we consider generalizations of $\Gamma(R)$ by modifying the vertices or adjacency relations of $\Gamma(R)$. In particular, we study the extended Zero-Divisor graph $\overline{\Gamma}(R)$, the annihilator graph $AG(R)$, and their analogs for ideal-based and congruence-based graphs.
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The Zero-Divisor graph of a commutative ring without identity
International Electronic Journal of Algebra, 2018Co-Authors: David F. Anderson, Darrin WeberAbstract:Let R be a commutative ring. The Zero-Divisor graph of R is the (simple) graph
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A GENERAL THEORY OF Zero-Divisor GRAPHS OVER A COMMUTATIVE RING
International Electronic Journal of Algebra, 2016Co-Authors: David F. Anderson, Elizabeth Fowler LewisAbstract:Let R be a commutative ring with 1 6= 0, I a proper ideal of R, and ∼ a multiplicative congruence relation on R. Let R/∼ = { [x]∼ | x ∈ R } be the commutative monoid of ∼-congruence classes under the induced multiplication [x]∼[y]∼ = [xy]∼, and let Z(R/∼) be the set of Zero-Divisors of R/∼. The ∼-Zero-Divisor graph of R is the (simple) graph Γ∼(R) with vertices Z(R/∼) \{[0]∼} and with distinct vertices [x]∼ and [y]∼ adjacent if and only if [x]∼[y]∼ = [0]∼. Special cases include the usual Zero-Divisor graphs Γ(R) and Γ(R/I), the ideal-based Zero-Divisor graph ΓI (R), and the compressed Zero-Divisor graphs ΓE(R) and ΓE(R/I). In this paper, we investigate the structure and relationship between the various ∼-Zero-Divisor graphs.
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Zero-Divisor Labelings of Graphs
Communications in Algebra, 2016Co-Authors: Pranjali, David F. Anderson, B. D. Acharya, Purnima GuptaAbstract:This paper introduces the notions of a Zero-Divisor labeling and the Zero-Divisor index of a graph using the Zero-Divisors of a commutative ring. Viewed in this way, the usual Zero-Divisor graph is a maximal graph with respect to a Zero-Divisor labeling. We also study optimal Zero-Divisor labelings of a finite graph.
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some remarks on the compressed Zero Divisor graph
Journal of Algebra, 2016Co-Authors: David F. Anderson, John D. LagrangeAbstract:Abstract Let R be a commutative ring with 1 ≠ 0 . The Zero-Divisor graph Γ ( R ) of R is the (undirected) graph with vertices the nonZero Zero-Divisors of R, and distinct vertices r and s are adjacent if and only if r s = 0 . The relation on R given by r ∼ s if and only if ann R ( r ) = ann R ( s ) is an equivalence relation. The compressed Zero-Divisor graph Γ E ( R ) of R is the (undirected) graph with vertices the equivalence classes induced by ∼ other than [0] and [1], and distinct vertices [ r ] and [ s ] are adjacent if and only if r s = 0 . Let R E be the set of equivalence classes for ∼ on R. Then R E is a commutative monoid with multiplication [ r ] [ s ] = [ r s ] . In this paper, we continue our study of the monoid R E and the compressed Zero-Divisor graph Γ E ( R ) . We consider several equivalence relations on R and their corresponding graph-theoretic translations to Γ ( R ) . We also show that the girth of Γ E ( R ) is three if it contains a cycle and determine the structure of Γ E ( R ) when it is acyclic and the monoids R E when Γ E ( R ) is a star graph.
Abdollah Alhevaz - One of the best experts on this subject based on the ideXlab platform.
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On diameter of the Zero-Divisor and the compressed Zero-Divisor graphs of skew Laurent polynomial rings
Journal of Algebra and Its Applications, 2019Co-Authors: Ebrahim Hashemi, Mona Abdi, Abdollah AlhevazAbstract:Let [Formula: see text] be an associative ring with nonZero identity. The Zero-Divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the nonZero Zero-Divisors of [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the set of all right annihilators and the set of all left annihilator of an element [Formula: see text], respectively, and let [Formula: see text]. The relation on [Formula: see text] given by [Formula: see text] if and only if [Formula: see text] is an equivalence relation. The compressed Zero-Divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the equivalence classes induced by [Formula: see text] other than [Formula: see text] and [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. The goal of our paper is to study the diameter of Zero-Divisor and the compressed Zero-Divisor graph of skew Laurent polynomial rings over noncommutative rings. We give a complete characterization of the possible diameters of [Formula: see text] and [Formula: see text], where the base ring [Formula: see text] is reversible and also has the [Formula: see text]-compatible property.
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on diameter of the Zero Divisor and the compressed Zero Divisor graphs of skew laurent polynomial rings
Journal of Algebra and Its Applications, 2019Co-Authors: Ebrahim Hashemi, Mona Abdi, Abdollah AlhevazAbstract:Let R be an associative ring with nonZero identity. The Zero-Divisor graph Γ(R) of R is the (undirected) graph with vertices the nonZero Zero-Divisors of R, and distinct vertices a and b are adjace...
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on Zero Divisor graphs of skew polynomial rings over non commutative rings
Journal of Algebra and Its Applications, 2017Co-Authors: Ebrahim Hashemi, R Amirjan, Abdollah AlhevazAbstract:In this paper, we continue to study Zero-Divisor properties of skew polynomial rings R[x; α,δ], where R is an associative ring equipped with an endomorphism α and an α-derivation δ. For an associative ring R, the undirected Zero-Divisor graph of R is the graph Γ(R) such that the vertices of Γ(R) are all the nonZero Zero-Divisors of R and two distinct vertices x and y are connected by an edge if and only if xy = 0 or yx = 0. As an application of reversible rings, we investigate the interplay between the ring-theoretical properties of a skew polynomial ring R[x; α,δ] and the graph-theoretical properties of its Zero-Divisor graph Γ(R[x; α,δ]). Our goal in this paper is to give a characterization of the possible diameters of (Γ(R[x; α,δ])) in terms of the diameter of Γ(R), when the base ring R is reversible and also have the (α,δ)-compatible property. We also completely describe the associative rings all of whose Zero-Divisor graphs of skew polynomials are complete.
K. Khashyarmanesh - One of the best experts on this subject based on the ideXlab platform.
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The Zero-Divisor Graph of a Lattice
Results in Mathematics, 2012Co-Authors: E. Estaji, K. KhashyarmaneshAbstract:For a finite bounded lattice £ , we associate a Zero-Divisor graph G ( £ ) which is a natural generalization of the concept of Zero-Divisor graph for a Boolean algebra. Also, we study the interplay of lattice-theoretic properties of £ with graph-theoretic properties of G ( £ ).
