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John D. Lagrange  One of the best experts on this subject based on the ideXlab platform.

some remarks on the compressed Zero Divisor graph
Journal of Algebra, 2016CoAuthors: David F. Anderson, John D. LagrangeAbstract:Abstract Let R be a commutative ring with 1 ≠ 0 . The ZeroDivisor graph Γ ( R ) of R is the (undirected) graph with vertices the nonZero ZeroDivisors 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 ZeroDivisor 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 ZeroDivisor graph Γ E ( R ) . We consider several equivalence relations on R and their corresponding graphtheoretic 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.

characterizations of three classes of Zero Divisor graphs
Canadian Mathematical Bulletin, 2012CoAuthors: John D. LagrangeAbstract:The ZeroDivisor graph Γ(R) of a commutative ring R is the graph whose vertices consist of the nonZero ZeroDivisors 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 ZeroDivisor graphs of Boolean rings. Also, commutative rings R such that Γ(R) is isomorphic to the ZeroDivisor graph of a direct product of integral domains are classified, as well as those whose ZeroDivisor graphs are central vertex complete. School of Natural Sciences, Indiana University Southeast, New Albany, Indiana 47150, USA email: 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

On Realizing ZeroDivisor Graphs
Communications in Algebra, 2008CoAuthors: John D. LagrangeAbstract:An algorithm is presented for constructing the ZeroDivisor graph of a direct product of integral domains. Moreover, graphs which are realizable as ZeroDivisor graphs of direct products of integral domains are classified, as well as those of Boolean rings. In particular, graphs which are realizable as ZeroDivisor graphs of finite reduced commutative rings are classified.

Complemented ZeroDivisor graphs and Boolean rings
Journal of Algebra, 2007CoAuthors: John D. LagrangeAbstract:Abstract For a commutative ring R, the ZeroDivisor graph of R is the graph whose vertices are the nonZero ZeroDivisors of R such that the vertices x and y are adjacent if and only if x y = 0 . In this paper, we classify the ZeroDivisor 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 ZeroDivisor 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.

ZeroDivisor graphs and total coloring conjecture
Soft Computing, 2020CoAuthors: Nilesh Khandekar, Vinayak JoshiAbstract:In this paper, we prove that the ZeroDivisor graphs of a special class of pseudocomplemented posets satisfy the total coloring conjecture. Also, we determine the edge chromatic number of the ZeroDivisor graphs of this special class of pseudocomplemented posets. These results are applied to ZeroDivisor graphs of finite reduced commutative rings.

ZeroDivisor graphs of lower dismantlable lattices II
Mathematica Slovaca, 2018CoAuthors: Avinash Patil, B. N. Waphare, Vinayak JoshiAbstract:Abstract In this paper, we continue our study of the ZeroDivisor graphs of lower dismantlable lattices that was started in [PATIL, A.—WAPHARE, B. N.—JOSHI, V.—POURALI, H. Y.: ZeroDivisor graphs of lower dismantlable lattices I, Math. Slovaca 67 (2017), 285–296]. The present paper mainly deals with an Isomorphism Problem for the ZeroDivisor graphs of lattices. In fact, we prove that the ZeroDivisor graphs of lower dismantlable lattices with the greatest element 1 as joinreducible are isomorphic if and only if the lattices are isomorphic.

perfect Zero Divisor graphs
Discrete Mathematics, 2017CoAuthors: Avinash Patil, B. N. Waphare, Vinayak JoshiAbstract:In this article, we characterize various algebraic and order structures whose ZeroDivisor 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).

the Zero Divisor graphs of boolean posets
Mathematica Slovaca, 2014CoAuthors: 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.

On generalized Zero Divisor graph of a poset
Discrete Applied Mathematics, 2013CoAuthors: 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 rpartite 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 semicomplemented 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.

GENERALIZATIONS OF THE ZeroDivisor GRAPH
International Electronic Journal of Algebra, 2020CoAuthors: David F. Anderson, Grace McclurkinAbstract:Let $R$ be a commutative ring with $1 \neq 0$ and $Z(R)$ its set of ZeroDivisors. The ZeroDivisor 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 ZeroDivisor graph $\overline{\Gamma}(R)$, the annihilator graph $AG(R)$, and their analogs for idealbased and congruencebased graphs.

The ZeroDivisor graph of a commutative ring without identity
International Electronic Journal of Algebra, 2018CoAuthors: David F. Anderson, Darrin WeberAbstract:Let R be a commutative ring. The ZeroDivisor graph of R is the (simple) graph

A GENERAL THEORY OF ZeroDivisor GRAPHS OVER A COMMUTATIVE RING
International Electronic Journal of Algebra, 2016CoAuthors: 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 ZeroDivisors of R/∼. The ∼ZeroDivisor 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 ZeroDivisor graphs Γ(R) and Γ(R/I), the idealbased ZeroDivisor graph ΓI (R), and the compressed ZeroDivisor graphs ΓE(R) and ΓE(R/I). In this paper, we investigate the structure and relationship between the various ∼ZeroDivisor graphs.

ZeroDivisor Labelings of Graphs
Communications in Algebra, 2016CoAuthors: Pranjali, David F. Anderson, B. D. Acharya, Purnima GuptaAbstract:This paper introduces the notions of a ZeroDivisor labeling and the ZeroDivisor index of a graph using the ZeroDivisors of a commutative ring. Viewed in this way, the usual ZeroDivisor graph is a maximal graph with respect to a ZeroDivisor labeling. We also study optimal ZeroDivisor labelings of a finite graph.

some remarks on the compressed Zero Divisor graph
Journal of Algebra, 2016CoAuthors: David F. Anderson, John D. LagrangeAbstract:Abstract Let R be a commutative ring with 1 ≠ 0 . The ZeroDivisor graph Γ ( R ) of R is the (undirected) graph with vertices the nonZero ZeroDivisors 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 ZeroDivisor 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 ZeroDivisor graph Γ E ( R ) . We consider several equivalence relations on R and their corresponding graphtheoretic 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.

On diameter of the ZeroDivisor and the compressed ZeroDivisor graphs of skew Laurent polynomial rings
Journal of Algebra and Its Applications, 2019CoAuthors: Ebrahim Hashemi, Mona Abdi, Abdollah AlhevazAbstract:Let [Formula: see text] be an associative ring with nonZero identity. The ZeroDivisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the nonZero ZeroDivisors 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 ZeroDivisor 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 ZeroDivisor and the compressed ZeroDivisor 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.

on diameter of the Zero Divisor and the compressed Zero Divisor graphs of skew laurent polynomial rings
Journal of Algebra and Its Applications, 2019CoAuthors: Ebrahim Hashemi, Mona Abdi, Abdollah AlhevazAbstract:Let R be an associative ring with nonZero identity. The ZeroDivisor graph Γ(R) of R is the (undirected) graph with vertices the nonZero ZeroDivisors of R, and distinct vertices a and b are adjace...

on Zero Divisor graphs of skew polynomial rings over non commutative rings
Journal of Algebra and Its Applications, 2017CoAuthors: Ebrahim Hashemi, R Amirjan, Abdollah AlhevazAbstract:In this paper, we continue to study ZeroDivisor properties of skew polynomial rings R[x; α,δ], where R is an associative ring equipped with an endomorphism α and an αderivation δ. For an associative ring R, the undirected ZeroDivisor graph of R is the graph Γ(R) such that the vertices of Γ(R) are all the nonZero ZeroDivisors 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 ringtheoretical properties of a skew polynomial ring R[x; α,δ] and the graphtheoretical properties of its ZeroDivisor 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 ZeroDivisor graphs of skew polynomials are complete.
K. Khashyarmanesh  One of the best experts on this subject based on the ideXlab platform.

The ZeroDivisor Graph of a Lattice
Results in Mathematics, 2012CoAuthors: E. Estaji, K. KhashyarmaneshAbstract:For a finite bounded lattice £ , we associate a ZeroDivisor graph G ( £ ) which is a natural generalization of the concept of ZeroDivisor graph for a Boolean algebra. Also, we study the interplay of latticetheoretic properties of £ with graphtheoretic properties of G ( £ ).