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S Akbari - One of the best experts on this subject based on the ideXlab platform.
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the proof of a conjecture in jacobson graph of a Commutative Ring
Journal of Algebra and Its Applications, 2015Co-Authors: S Akbari, S Khojasteh, A YousefzadehfardAbstract:Let R be a Commutative Ring with nonzero identity. The Jacobson graph of R denoted by 𝔍R is a graph with the vertex set R\J(R), and two distinct vertices x, y ∈ V(𝔍R) are adjacent if and only if 1 - xy ∉ U(R), where U(R) is the set of all unit elements of R. Let ω(𝔍R) denote the clique number of 𝔍R. It was conjectured that if is a Commutative finite Ring and (Ri, 𝔪i) is a local Ring, for i = 1, …, n, then , where Fi = Ri/𝔪i, for i = 1, …, n. In this paper, we prove that if R is a Commutative Ring (not necessarily finite) and R is not a field, then ω(𝔍R) = max𝔪∈Max(R) |𝔪| and using this we show that the aforementioned conjecture holds.
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some properties of a cayley graph of a Commutative Ring
Communications in Algebra, 2014Co-Authors: Ghodratollah Aalipour, S AkbariAbstract:Let R be a Commutative Ring with unity and R +, U(R), and Z*(R) be the additive group, the set of unit elements, and the set of all nonzero zero-divisors of R, respectively. We denote by ℂ𝔸𝕐(R) and G R , the Cayley graph Cay(R +, Z*(R)) and the unitary Cayley graph Cay(R +, U(R)), respectively. For an Artinian Ring R, Akhtar et al. (2009) studied G R . In this article, we study ℂ𝔸𝕐(R) and determine the clique number, chromatic number, edge chromatic number, domination number, and the girth of ℂ𝔸𝕐(R). We also characterize all Rings R whose ℂ𝔸𝕐(R) is planar. Moreover, we determine all finite Rings R whose ℂ𝔸𝕐(R) is strongly regular. We prove that ℂ𝔸𝕐(R) is strongly regular if and only if it is edge transitive. As a consequence, we characterize all finite Rings R for which G R is a strongly regular graph.
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the regular graph of a non Commutative Ring
Electronic Notes in Discrete Mathematics, 2014Co-Authors: S Akbari, F HeydariAbstract:Abstract Let R be a Ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by T ( Γ ( R ) ) is a graph with all elements of R as vertices, and two distinct vertices x , y ∈ R are adjacent if and only if x + y ∈ Z ( R ) . Let the regular graph of R , R e g ( Γ ( R ) ) , be the induced subgraph of T ( Γ ( R ) ) on the regular elements of R. In 2008, Anderson and Badawi proved that the girth of total graph and regular graph of a Commutative Ring are contained in the set { 3 , 4 , ∞ } . In this paper, we extend this result to an arbitrary Ring (not necessarily Commutative). Also, we prove that if R is a reduced left Noetherian Ring and 2 ∉ Z ( R ) , then the chromatic number and the clique number of R e g ( Γ ( R ) ) are the same and they are 2 r , where r is the number of minimal prime ideals of R. Among other results we show that if R is a semiprime left Noetherian Ring and R e g ( R ) is finite, then R is finite.
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the regular graph of a Commutative Ring
Periodica Mathematica Hungarica, 2013Co-Authors: S Akbari, F HeydariAbstract:Let R be a Commutative Ring, let Z(R) be the set of all zero-divisors of R and Reg(R) = R\Z(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+y ∈ Z(R). In this paper we show that if R is a Commutative Noetherian Ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2n, where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a Ring have at most two vertices.
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on the cayley graph of a Commutative Ring with respect to its zero divisors
arXiv: Combinatorics, 2013Co-Authors: Ghodratollah Aalipour, S AkbariAbstract:Let $R$ be a Commutative Ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In this paper, we study $\mathbb{CAY}(R)$. Among other results, it is shown that for every zero-dimensional non-local Ring $R$, $\mathbb{CAY}(R)$ is a connected graph of diameter 2. Moreover, for a finite Ring $R$, we obtain the vertex connectivity and the edge connectivity of $\mathbb{CAY}(R)$. We investigate Rings $R$ with perfect $\mathbb{CAY}(R)$ as well. We also study $Reg(\mathbb{CAY}(R))$ the induced subgraph on the regular elements of $R$. This graph gives a family of vertex transitive graphs. We show that if $R$ is a Noetherian Ring and $Reg(\mathbb{CAY}(R))$ has no infinite clique, then $R$ is finite. Furthermore, for every finite Ring $R$, the clique number and the chromatic number of $Reg(\mathbb{CAY}(R))$ are determined.
Kazem Khashyarmanesh - One of the best experts on this subject based on the ideXlab platform.
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The annihilator ideal graph of a Commutative Ring
Note di Matematica, 2016Co-Authors: M. Afkhami, Nesa Hoseini, Kazem KhashyarmaneshAbstract:Let R be a Commutative Ring with nonzero identity and I be a proper ideal of R. The annihilator graph of R with respect to I, which is denoted by AG I (R), is the undirected graph with vertex-set V (AG I (R) = (x ε(lunate) R\I: xy ε(lunate) I for some y ε(lunate) I) and two distinct vertices x and y are adjacent if and only if A I (xy) ≠ A I (x) U A I (y), where A I (x) = (r ε(lunate) R: rx ε(lunate) I). In this paper, we study some basic properties of AG I (R), and we characterise when AG I (R) is planar, outerplanar or a Ring graph. Also, we study the graph AG I (Zn), where Zn is the Ring of integers modulo n.
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cayley graphs of ideals in a Commutative Ring
Bulletin of the Malaysian Mathematical Sciences Society, 2014Co-Authors: M. Afkhami, M R A Hmadi, R Ahani J N Ezhad, Kazem KhashyarmaneshAbstract:Let R be a Commutative Ring. We associate a digraph to the ideals of R whose vertex set is the set of all nontrivial ideals of R and, for every two distinct vertices I and J , there is an arc from I to J , denoted by I → J , whenever there exists a nontrivial ideal L such that J = IL. We call this graph the ideal digraph of R and denote it by −→ IΓ(R). Also, for a semigroup H and a subset S of H, the Cayley graph Cay(H,S) of H relative to S is defined as the digraph with vertex set H and edge set E(H,S) consisting of those ordered pairs (x, y) such that y = sx for some s ∈ S. In fact the ideal digraph −→ IΓ(R) is isomorphic to the Cayley graph Cay(I, I), where I is the set of all ideals of R and I consists of nontrivial ideals. The undirected ideal (simple) graph of R, denoted by IΓ(R), has an edge joining I and J whenever either J = IL or I = JL, for some nontrivial ideal L of R. In this paper, we study some basic properties of graphs −→ IΓ(R) and IΓ(R) such as connectivity, diameter, graph height, Wiener index and clique number. Moreover, we study the Hasse ideal digraph −→ HΓ(R), which is a spanning subgraph of −→ IΓ(R) such that for each two distinct vertices I and J , there is an arc from I to J in −→ HΓ(R) whenever I → J in −→ IΓ(R), and there is no vertex L such that I → L and L → J in −→ IΓ(R).
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planar outerplanar and Ring graph of the cozero divisor graph of a finite Commutative Ring
Journal of Algebra and Its Applications, 2012Co-Authors: M. Afkhami, Kazem KhashyarmaneshAbstract:Let R be a Commutative Ring with nonzero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex-set W*(R), which is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b in W*(R) are adjacent if and only if a ∉ Rb and b ∉ Ra. In this paper, we characterize all finite Commutative Rings R such that Γ′(R) is planar, outerplanar or Ring graph.
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on the associated graphs to a Commutative Ring
Journal of Algebra and Its Applications, 2012Co-Authors: Zahra Barati, Kazem Khashyarmanesh, Fatemeh Mohammadi, Kh NafarAbstract:Let R be a Commutative Ring with nonzero identity. For an arbitrary multiplicatively closed subset S of R, we associate a simple graph denoted by ΓS(R) with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x+y ∈ S. Two well-known graphs of this type are the total graph and the unit graph. In this paper, we study some basic properties of ΓS(R). Moreover, we will improve and generalize some results for the total and the unit graphs.
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a generalization of the unit and unitary cayley graphs of a Commutative Ring
Acta Mathematica Hungarica, 2012Co-Authors: Kazem Khashyarmanesh, Mahdi Reza KhorsandiAbstract:Let R be a Commutative Ring with non-zero identity and G be a multiplicative subgroup of U(R), where U(R) is the multiplicative group of unit elements of R. Also, suppose that S is a non-empty subset of G such that S−1={s−1∣s∈S}⫅S. Then we define Γ(R,G,S) to be the graph with vertex set R and two distinct elements x,y∈R are adjacent if and only if there exists s∈S such that x+sy∈G. This graph provides a generalization of the unit and unitary Cayley graphs. In fact, Γ(R,U(R),S) is the unit graph or the unitary Cayley graph, whenever S={1} or S={−1}, respectively. In this paper, we study the properties of the graph Γ(R,G,S) and extend some results in the unit and unitary Cayley graphs.
Jafar Amjadi - One of the best experts on this subject based on the ideXlab platform.
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The essential ideal graph of a Commutative Ring
Asian-European Journal of Mathematics, 2020Co-Authors: Jafar AmjadiAbstract:Let [Formula: see text] be a Commutative Ring with identity. The essential ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] is an essential ideal. In this paper, we initiate the study of the essential ideal graph of a Commutative Ring and we investigate its properties.
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The co-annihilating graph of a Commutative Ring
Discrete Mathematics Algorithms and Applications, 2018Co-Authors: Jafar Amjadi, Abbas AlilouAbstract:Let [Formula: see text] be a Commutative Ring with identity and [Formula: see text] be the set of all non-zero non-units of [Formula: see text]. The co-annihilating graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertex set [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text]. In this paper, we initiate the study of the co-annihilating graph of a Commutative Ring and we investigate its properties.
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The co-annihilating graph of a Commutative Ring
Discrete Mathematics Algorithms and Applications, 2017Co-Authors: Jafar Amjadi, Abbas AlilouAbstract:Let R be a Commutative Ring with identity and 𝔘R be the set of all non-zero non-units of R. The co-annihilating graph of R, denoted by 𝒞𝒜R, is a graph with vertex set 𝔘R and two vertices a and b are adjacent whenever Ann(a) ∩Ann(b) = (0). In this paper, we initiate the study of the co-annihilating graph of a Commutative Ring and we investigate its properties.
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The essential ideal graph of a Commutative Ring
Asian-european Journal of Mathematics, 2017Co-Authors: Jafar AmjadiAbstract:Let R be a Commutative Ring with identity. The essential ideal graph of R, denoted by ℰR, is a graph whose vertex set is the set of all nonzero proper ideals of R and two vertices I and J are adjacent whenever I + J is an essential ideal. In this paper, we initiate the study of the essential ideal graph of a Commutative Ring and we investigate its properties.
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a new graph associated to a Commutative Ring
Discrete Mathematics Algorithms and Applications, 2016Co-Authors: Abbas Alilou, Jafar Amjadi, Seyed Mahmoud SheikholeslamiAbstract:Let R be a Commutative Ring with identity. In this paper, we consider a simple graph associated with R denoted by ΩR∗, whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we initiate the study of the graph ΩR∗ and we investigate its properties. In particular, we show that ΩR∗ is a connected graph with diam(ΩR∗) ≤ 3 unless R is isomorphic to a direct product of two fields. Moreover, we characterize all Commutative Rings R with at least two maximal ideals for which ΩR∗ are planar.
M J Nikmehr - One of the best experts on this subject based on the ideXlab platform.
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on the essential graph of a Commutative Ring
Journal of Algebra and Its Applications, 2017Co-Authors: M J Nikmehr, Reza Nikandish, M BakhtyiariAbstract:Let R be a Commutative Ring with identity, and let Z(R) be the set of zero-divisors of R. The essential graph of R is defined as the graph EG(R) with the vertex set Z(R)∗ = Z(R)\{0}, and two distinct vertices x and y are adjacent if and only if annR(xy) is an essential ideal. It is proved that EG(R) is connected with diameter at most three and with girth at most four, if EG(R) contains a cycle. Furthermore, Rings with complete or star essential graphs are characterized. Also, we study the affinity between essential graph and zero-divisor graph that is associated with a Ring. Finally, we show that the essential graph associated with an Artinian Ring is weakly perfect, i.e. its vertex chromatic number equals its clique number.
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coloRing of the annihilator graph of a Commutative Ring
Journal of Algebra and Its Applications, 2016Co-Authors: Reza Nikandish, M J Nikmehr, M BakhtyiariAbstract:Let R be a Commutative Ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is defined as the graph AG(R) with the vertex set Z(R)∗ = Z(R)\{0}, and two distinct vertices x and y are adjacent if and only if annR(xy)≠annR(x) ∪annR(y). In this paper, we study annihilator graphs of Rings with equal clique number and chromatic number. For some classes of Rings, we give an explicit formula for the clique number of annihilator graphs. Among other results, bipartite annihilator graphs of Rings are characterized. Furthermore, some results on annihilator graphs with finite clique number are given.
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the m principal graph of a Commutative Ring
Periodica Mathematica Hungarica, 2014Co-Authors: M J Nikmehr, F HeydariAbstract:Let \(R\) be a Commutative Ring and \(M\) be an \(R\)-module. In this paper, we introduce the \(M\)-principal graph of \(R\), denoted by \(M-PG(R)\). It is the graph whose vertex set is \(R\backslash \{0\}\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xM=yM\). In the special case that \(M=R, M-PG(R)\) is denoted by \(PG(R)\). The basic properties and possible structures of these two graphs are studied. Also, some relations between \(PG(R)\) and \(M-PG(R)\) are established.
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the m regular graph of a Commutative Ring
arXiv: Commutative Algebra, 2013Co-Authors: M J Nikmehr, F HeydariAbstract:Let $R$ be a Commutative Ring and $M$ be an $R$-module, and let $Z(M)$ be the set of all zero-divisors on $M$. In 2008, D.F. Anderson and A. Badawi introduced the regular graph of $R$. In this paper, we generalize the regular graph of $R$ to the \textit{$M$-regular graph} of $R$, denoted by $M$-$Reg(\Gamma(R))$. It is the undirected graph with all $M$-regular elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(M)$. The basic properties and possible structures of the $M$-$Reg(\Gamma(R))$ are studied. We determine the girth of the $M$-regular graph of $R$. Also, we provide some lower bounds for the independence number and the clique number of the $M$-$Reg(\Gamma(R))$. Among other results, we prove that for every Noetherian Ring $R$ and every finitely generated module $M$ over $R$, if $2\notin Z(M)$ and the independence number of the $M$-$Reg(\Gamma(R))$ is finite, then $R$ is finite.
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on the coloRing of the annihilating ideal graph of a Commutative Ring
Discrete Mathematics, 2012Co-Authors: Ghodratollah Aalipour, S Akbari, M J Nikmehr, Reza Nikandish, Farzad ShaveisiAbstract:Abstract Suppose that R is a Commutative Ring with identity. Let A ( R ) be the set of all ideals of R with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph A G ( R ) with the vertex set A ( R ) ∗ = A ( R ) ∖ { ( 0 ) } and two distinct vertices I and J are adjacent if and only if I J = ( 0 ) . In Behboodi and Rakeei (2011) [8] , it was conjectured that for a reduced Ring R with more than two minimal prime ideals, g i r t h ( A G ( R ) ) = 3 . Here, we prove that for every (not necessarily reduced) Ring R , ω ( A G ( R ) ) ≥ | Min ( R ) | , which shows that the conjecture is true. Also in this paper, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a Commutative Ring. Among other results, it is shown that if the chromatic number of the zero-divisor graph is finite, then the chromatic number of the annihilating-ideal graph is finite too. We investigate Commutative Rings whose annihilating-ideal graphs are bipartite. It is proved that A G ( R ) is bipartite if and only if A G ( R ) is triangle-free.
T Asir - One of the best experts on this subject based on the ideXlab platform.
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on the genus of generalized unit and unitary cayley graphs of a Commutative Ring
Acta Mathematica Hungarica, 2014Co-Authors: T Asir, Tamizh T ChelvamAbstract:Let R be a Commutative Ring, U (R) be the set of all unit el- ements of R, G be a multiplicative subgroup of U (R )a ndS be a non-empty subset of G such that S −1 = {s −1 : s ∈ S} S. In (16), K. Khashyarmanesh et al. defined a graph of R, denoted by Γ(R, G, S), which generalizes both unit and unitary Cayley graphs of R. In this paper, we derive several bounds for the genus of Γ(R, U (R) ,S ). Moreover, we characterize all Commutative Artinian Rings R for which the genus of Γ(R, U (R) ,S ) is one. This leads to the characterization of all Commutative Artinian Rings whose unit and unitary Cayley graphs have genus one.
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on the total graph and its complement of a Commutative Ring
Communications in Algebra, 2013Co-Authors: T Asir, T. Tamizh ChelvamAbstract:Let R be a Commutative Ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by T Γ(R), is the undirected graph with vertex set R and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). denotes the complement of T Γ(R). The study on total graphs has been initiated by D. F. Anderson and A. Badawi [2]. In this article, we characterize all Commutative Rings whose total graph (or its complement) is in some known class of graphs. Also we determine the structure whenever |Reg(R)| = 2. Further, we obtain certain necessary conditions for to be connected whenever is connected and prove that . It is also proved that if diam(T Γ(R)) = 2, then T Γ(R) is Hamiltonian, which is a generalization of a characterization proved by S. Akbari et al. [1].
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the intersection graph of gamma sets in the total graph of a Commutative Ring i
Journal of Algebra and Its Applications, 2013Co-Authors: T. Tamizh Chelvam, T AsirAbstract:Let R be a Commutative Ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a Commutative Ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R, denoted by TΓ(R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a Commutative Ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of TΓ(R) where R is a Commutative Artin Ring. The intersection graph of gamma sets in TΓ(R) is denoted by ITΓ(R). Tamizh Chelvam and Asir [Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory32 (2012) 339–354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph ITΓ (ℤn) of gamma sets in TΓ(ℤn). In this paper, we study about ITΓ(R), where R is a Commutative Artin Ring. Actually we investigate the interplay between graph-theoretic properties of ITΓ(R) and Ring-theoretic properties of R. At the first instance, we prove that diam(ITΓ(R)) ≤ 2 and gr(ITΓ(R)) ≤ 4. Also some characterization results regarding completeness, bipartite, cycle and chordal nature of ITΓ(R) are given. Further, we discuss about the vertex-transitive property of ITΓ(R). At last, we obtain all Commutative Artin Rings R for which ITΓ(R) is either planar or toroidal or genus two.
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on the genus of the total graph of a Commutative Ring
Communications in Algebra, 2013Co-Authors: T. Tamizh Chelvam, T AsirAbstract:Let R be a Commutative Ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by TΓ(R), is the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Maimani et al. [13] determined all isomorphism classes of finite Commutative Rings whose total graph has genus at most one. In this article, after enumerating certain lower and upper bounds for genus of the total graph of a Commutative Ring, we characterize all isomorphism classes of finite Commutative Rings whose total graph has genus two.
