Annihilator Graph Research Articles

On the annihilator graphs of partial transformation semigroups

Let X n = { 1 , 2 , … , n } and P n be a partial transformation semigroup on X n . Obviously, the empty set ∅ is a zero element of P n and denoted by 0. Let P n * = P n \\ { 0 } . An element α ∈ P n is a zero divisor of P n if there exists β ∈ P n * such that α β = 0 = β α . The set Ann ( α ) = { β ∈ P n : α β = 0 = β α } is called the annihilator of α in P n . It is clear that 0 ∈ Ann ( α ) for all α ∈ P n . Let Z ( P n ) be the set of all zero divisors of P n and Z ( P n ) * = Z ( P n ) \\ { 0 } . Generally, if α ∈ Z ( P n ) , then there exists β ∈ Z ( P n ) * in which β ∈ Ann ( α ) . In this paper, we construct the annihilator graph Γ n of P n which is an undirected simple graph with vertex set Z ( P n ) * and two distinct vertices α and β are adjacent if and only if Ann ( α ) ∩ Ann ( β ) ≠ { 0 } . Furthermore, we prove some basic structural properties of Γ n and determine invariants for Γ n such as the diameter, girth, clique, domination number, and independence number.

Annihilator graphs derived from group rings

Let RG be the group ring of the group G over a ring R and let Z∗(RG) be the collection of all non-zero zero divisors in a finite group ring RG. And, for x ∈ Z∗(RG), ann(x) = { r ∈ RG | rx = 0}. The annihilator graph of RG denoted as AG(RG), and is defined as the graph whose vertex set is the elements of non-zero zero divisors in RG and two distinct vertices vx and vy are adjacent if and only if ann(x) ∩ ann(y) ≠ 0. In this paper we try to characterize the properties of annihilator graphs in group rings.

Twin-free cliques in annihilator graphs of commutative rings

For a connected graph [Formula: see text] a clique [Formula: see text] is twin-free if every pair of elements of [Formula: see text] have distinct closed neighborhoods and the number of elements in a twin-free clique of maximum cardinality is called twin-free clique number of [Formula: see text]. The annihilator graph [Formula: see text] of a commutative and unital ring [Formula: see text] is a graph whose vertices are all non-zero zero-divisors of [Formula: see text] and there is an edge between two distinct vertices [Formula: see text] if and only if [Formula: see text] is properly contained in [Formula: see text]. In this paper, twin-free clique number of [Formula: see text] is computed and as an application the strong metric dimension of [Formula: see text] is characterized. Among other things, for a reduced ring [Formula: see text], the forcing strong metric dimension of [Formula: see text] is given.

Annihilator graphs of a commutative semigroup whose Zero-divisor graphs are a complete graph with an end vertex

Abstract Suppose that the zero-divisor graph of a commutative semi-group S, be a complete graph with an end vertex. In this paper, we determine the structure of the annihilator graph S and we show that if Z(S)= S, then the annihilator graph S is a disconnected graph.

On torsion elements and their annihilators

Let R be a commutative ring with identity, and let M be an R-module. In this paper, we focus on the ideals of R that are annihilators of torsion elements of M. In analogy with definitions and results on zero-divisor and annihilator graphs of rings, we define the annihilator graph of a module. We investigate the structure, the diameter, and the girth of this graph and the closely related torsion graph of a module introduced by Ghalandarzadeh and Malakooti Rad. Given the right definitions, the properties of the modules are reflected in the graph theoretic properties of the graphs. We thus modify and extend results on zero divisors of rings to the much more general setting of modules and their torsion elements. In addition, we significantly strengthen the known results on the torsion graphs of modules. Some of the results will be refined further for the cases when M is a multiplication module or a reduced module.

Crosscap two of class of graphs from commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The annihilator graph of commutative ring [Formula: see text] is the simple undirected graph AG([Formula: see text]) with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal. In this paper, we classify all finite commutative rings with identity whose annihilator graph and essential graph have crosscap two.

On the annihilator graph of a commutative ring

On the annihilator graph of a commutative ring

Metric dimension of complement of annihilator graphs associated with commutative rings

Metric dimension of complement of annihilator graphs associated with commutative rings

THE ANNIHILATOR GRAPH FOR MODULED OVER COMMUTATIVE RINGS

‎Let $R$ be a commutative ring and $M$ be an $R$-module‎. ‎The‎ ‎annihilator graph of $M$‎, ‎denoted by $AG(M)$ is a simple undirected‎ ‎graph associated to $M$ whose the set of vertices is‎ ‎$Z_R(M) setminus {rm Ann}_R(M)$ and two distinct vertices $x$ and‎ ‎$y$ are adjacent if and only if ${rm Ann}_M(xy)neq {rm‎ ‎Ann}_M(x) cup {rm Ann}_M(y)$‎. ‎In this paper‎, ‎we study the‎ ‎diameter and the girth of $AG(M)$ and we characterize all modules‎ ‎whose annihilator graph is complete‎. ‎Furthermore‎, ‎we look for the‎ ‎relationship between the annihilator graph of $M$ and its zero-divisor‎ ‎graph‎.

The annihilator graph of modules over commutative rings

Let $M$ be a module over a commutative ring $R$, $Z_{*}(M)$ be its set of weak zero-divisor elements, andif $min M$, then let $I_m=(Rm:_R M)={rin R : rMsubseteq Rm}$. The annihilator graph of $M$ is the (undirected) graph$AG(M)$ with vertices $tilde{Z_{*}}(M)=Z_{*}(M)setminus {0}$, and two distinct vertices $m$ and $n$ are adjacent if andonly if $(0:_R I_{m}I_{n}M)neq (0:_R m)cup (0:_R n)$. We show that $AG(M)$ is connected with diameter at most two and girth at mostfour. Also, we study some properties of the zero-divisor graph of reduced multiplication-like $R$-modules.

On the Strong Metric Dimension of Annihilator Graphs of Commutative Rings

For a connected graph G(V, E), a vertex $$w\in V(G)$$ strongly resolves two vertices $$u, v \in V(G)$$ if there exists a shortest $$u-w$$ path containing v or a shortest $$v-w$$ path containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. 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 a simple graph with the vertex set $$Z(R)^*=Z(R){\setminus }\{0\}$$ , and two distinct vertices x and y are adjacent if and only if $$ann_R(xy)\ne ann_R(x)\cup ann_R(y)$$ . In this paper, we study the strong metric dimension of annihilator graphs associated with commutative rings and some strong metric dimension formulae for annihilator graphs are given.

When the annihilator graph of a commutative ring is planar or toroidal?

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 undirected 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 ann_R(xy) \neq ann_R(x) \cup ann_R(y). In this paper, all rings whose annihilator graphs can be embedded on the plane or torus are classified.

Hosoya Polynomial, Wiener Index, Coloring and Planar of Annihilator Graph of Zn

Let R be a commutative ring with identity. We consider ΓB(R) an annihilator graph of the commutative ring R. In this paper, we find Hosoya polynomial, Wiener index, Coloring, and Planar annihilator graph of Zn denote ΓB(Zn) , with n= pm or n=pmq, where p, q are distinct prime numbers and m is an integer with m ≥ 1 .

On perfect annihilator graphs of commutative rings

Let [Formula: see text] be a commutative ring with identity, and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The annihilator graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study the perfectness of annihilator graphs of a vast range of rings. Indeed, it is shown that if [Formula: see text] is reduced with finitely many minimal primes or nonreduced, then [Formula: see text] is perfect.

Annihilator graphs of MV-algebras

In this paper, we introduce the annihilator graph [Formula: see text] of an MV-algebra [Formula: see text]. We show that [Formula: see text] contains the zero-divisor graph [Formula: see text] as a spanning subgraph. We then prove that [Formula: see text] if and only if [Formula: see text]. Moreover, we obtain that the girth [Formula: see text].

Some Remarks on the Annihilator Graph of a Commutative Ring

Let $R$ be a commutative ring with nonzero identity. The annihilator graph of $R$, denoted by $AG(R)$, is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\ann_R(xy)\neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$. We investigate the interplay between ring-theoretic properties of $R$ and graph-theoretic properties of $AG(R)$. We study the relation between two graphs $\Gamma(R)$ and $AG(R)$, where $R$ is a non-reduced commutative ring. Also, we completely characterize the rings whose annihilator graphs are complete.

Annihilator graph of semiring of matrices over Boolean semiring

The annihilator graph of a semiring S, denoted by AG(S), is the graph whose vertex set is the set of all nonzero zero-divisors of S. In commutative semiring S, two distinct vertices are adjacent if and only if ann(xy) ≠ ann(x) ∪ ann(y), where ann(x) = {s ∈ S|sx = 0}. Similarly in noncommutative semiring S, two distinct vertices are connected by an edge if and only if either l. ann(xy) ≠ l. ann(x) ∪ l. ann(y), l. ann(yx) ≠ l. ann(x) ∪ l. ann(y), r. ann(xy) ≠ r. ann(x) ∪ r. ann(y), or r. ann(yx) ≠ r. ann(x) ∪ r. ann(y) where l. ann(x) = {s ∈ S|sx = 0} and r. ann(x) = {s ∈ S|xs = 0}. In this paper we study the properties of the right annihilator and the left annihilator of semiring of matrices over Boolean semiring Mn (ℬ) and then use these results to determine the diameter of the graph AG(Mn (ℬ)).

GENERALIZATIONS OF THE ZERO-DIVISOR GRAPH

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.

Compressed intersection annihilator graph

Let R be a commutative ring with a non-zero identity. In this paper, we define a new graph, the compressed intersection annihilator graph, denoted by $IA(R)$, and investigate some of its theoretical properties and its relation with the structure of the ring. It is a generalization of the torsion graph $\Gamma_{R}(R)$. We study classes of rings for which the equivalence between the set of zero-divisors of $R$ being an ideal and the completeness of $IA(R)$ holds. We also study the relation between $\Gamma_{R}(R)$ and $IA(R)$. In addition, we show that if the compressed intersection annihilator graph of a ring $R$ is finite, then there exists a subring $S$ of $R$ such that $IA(S)\cong IA(R)$. Also, we show that the compressed intersection annihilator graph will never be a complete bipartite graph. Besides, we show that the graph $IA(R)$ with at least three vertices is connected and its diameter is less than or equal to three. Finally, we determine the properties of the graph in the cases when $R$ is the ring of integers modulo $n$, the direct product of integral domains, the direct product of Artinine local rings and the direct product of two rings such that one of them is not an integral domain.

On two extensions of the classical zero-divisor graph

The extended zero-divisor graph and the annihilator graph of a ring are two extensions of the classical zero-divisor graph. In this paper, we investigate the relation between these graphs. Relation between these graphs on some particular ring constructions is also given.