Annihilator Graph Research Articles

The metric dimension of annihilator graphs of commutative rings

Let [Formula: see text] be a commutative ring with nonzero identity. The annihilator graph of [Formula: see text], denoted by [Formula: see text], is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of [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 metric dimension of annihilator graphs associated with commutative rings and some metric dimension formulae for annihilator graphs are given.

Comparison of graphs associated to a commutative Artinian ring

Let R be a commutative ring with $$1\ne 0$$ and the additive group $$R^+$$ . Several graphs on R have been introduced by many authors, among zero-divisor graph $$\Gamma _1(R)$$ , co-maximal graph $$\Gamma _2(R)$$ , annihilator graph AG(R), total graph $$ T(\Gamma (R))$$ , cozero-divisors graph $$\Gamma _\mathrm{c}(R)$$ , equivalence classes graph $$\Gamma _\mathrm{E}(R)$$ and the Cayley graph $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when R is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if R has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results, we prove that for a commutative finite ring R with $$|\mathrm{Max}(R)|=n \ge 3$$ , $$ \Gamma _1(R) \simeq \Gamma _2(R)$$ if and only if $$R\simeq \mathbb {Z}^n_2$$ ; if and only if $$\Gamma _1(R) \simeq \Gamma _\mathrm{E}(R)$$ . Also the annihilator graph is identical to the cozero-divisor graph if and only if R is a Frobenius ring.

The annihilator graph of a 0-distributive lattice

‎‎In this article‎, ‎for a lattice $mathcal L$‎, ‎we define and investigate‎ ‎the annihilator graph $mathfrak {ag} (mathcal L)$ of $mathcal L$ which contains the zero-divisor graph of $mathcal L$ as a subgraph‎. ‎Also‎, ‎for a 0-distributive lattice $mathcal L$‎, ‎we study some properties of this graph such as regularity‎, ‎connectedness‎, ‎the diameter‎, ‎the girth and its domination number‎. ‎Moreover‎, ‎for a distributive lattice $mathcal L$ with $Z(mathcal L)neqlbrace 0rbrace$‎, ‎we show that $mathfrak {ag} (mathcal L) = Gamma(mathcal L)$ if and only if $mathcal L$ has exactly two minimal prime ideals‎. ‎Among other things‎, ‎we consider the annihilator graph $mathfrak {ag} (mathcal L)$ of the lattice $mathcal L=(mathcal D(n),|)$ containing all positive divisors of a non-prime natural number $n$ and we compute some invariants such as the domination number‎, ‎the clique number and the chromatic number of this graph‎. ‎Also‎, ‎for this lattice we investigate some special cases in which $mathfrak {ag} (mathcal D(n))$ or $Gamma(mathcal D(n))$ are planar‎, ‎Eulerian or Hamiltonian.

The annihilation graphs of commutator posets and lattices with respect to an ideal

As an extension of [Formula: see text] (the annihilation graph of the commutator poset (lattice) [Formula: see text] with respect to an element [Formula: see text]), we discuss when [Formula: see text] (the annihilation graph of the commutator poset (lattice) [Formula: see text] with respect to an ideal [Formula: see text]) is a complete bipartite ([Formula: see text]-partite) graph together with some of its other graph-theoretic properties. We investigate the interplay between some (order-) lattice-theoretic properties of [Formula: see text] and graph-theoretic properties of its associated graph [Formula: see text]. We provide some examples to show that some conditions are not superfluous assumptions. We prove and show by a counterexample that the class of lower sets of a commutator poset [Formula: see text] is properly contained in the class of [Formula: see text]-ideals of [Formula: see text] (i.e. multiplicatively absorptive ideals (sets) of [Formula: see text] that are defined by commutator operation).

The Compressed Annihilator Graph of a Commutative Ring

Let R be a commutative ring. In this paper, we introduce and study the compressed annihilator graph of R. The compressed annihilator graph of R is the graph AGE(R), whose vertices are equivalence classes of zero-divisors of R and two distinct vertices [x] and [y] are adjacent if and only if ann(x)∪ann(y) ⊂ ann(xy). For a reduced ring R, we show that compressed annihilator graph of R is identical to the compressed zero-divisor graph of R if and only if 0 is a 2-absorbing ideal of R. As a consequence, we show that an Artinian ring R is either local or reduced whenever 0 is a 2-absorbing ideal of R.

Commutative rings with genus two annihilator graphs

ABSTRACTLet R be a commutative ring and Z(R)* be its set of all nonzero zero-divisors. The annihilator graph of a commutative ring R is the simple undirected graph AG(R) with vertices Z(R)*, and two distinct vertices x and y are adjacent if and only if ann(xy)≠ann(x)∪ann(y). The notion of annihilator graph has been introduced and studied by Badawi [8]. In this paper, we classify the finite commutative rings whose AG(R) are projective. Also we determine all isomorphism classes of finite commutative rings with identity whose AG(R) has genus two.

The annihilation graphs of commutator posets and lattices with respect to an element

We propose a new, widely generalized context for the study of the zero-divisor type (annihilating-ideal) graphs, where the vertices of graphs are not elements/ideals of a commutative ring, but elements of an abstract ordered set [lattice] (imitating the lattice of ideals of a ring), equipped with a commutative (not necessarily associative) binary operation (imitating the product of ideals of a ring). We discuss, when [Formula: see text] (the annihilation graph of the commutator poset [lattice] [Formula: see text] with respect to an element [Formula: see text]) is a complete bipartite graph together with some of its other graph-theoretic properties. In contrast to the case of rings, we construct a commutator poset whose [Formula: see text] contains a cut-point. We provide some examples to show that some conditions are not superfluous assumptions. We also give some examples of a large class of lattices, such as the lattice of ideals of a commutative ring, the lattice of normal subgroups of a group, and the lattice of all congruences on an algebra in a variety (congruence modular variety) by using the commutators as the multiplicative binary operation on these lattices. This shows that how the commutator theory can define and unify many zero-divisor type graphs of different algebraic structures as a special case of this paper.

On annihilator graph of a finite commutative ring

‎The annihilator graph $AG(R)$ of a commutative ring $R$ is a simple undirected graph with the vertex set $Z(R)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $ neq $ $ann(xy)$‎. ‎In this paper we give the sufficient condition for a graph $AG(R)$ to be complete‎. ‎We characterize rings for which $AG(R)$ is a regular graph‎, ‎we show that $gamma (AG(R))in {1,2}$ and we also characterize the rings for which $AG(R)$ has a cut vertex‎. ‎Finally we find the clique number of a finite reduced ring and characterize the rings for which $AG(R)$ is a planar graph‎.

English

Let R be a commutative ring. The annihilator graph of R, denoted by AG(R), is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann R (xy) ≠ ann R (x) ∪ ann R (y), where for z ∈ R, ann R (z) = {r ∈ R: rz = 0}. In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1, 2 or 3. Also, we investigate some properties of the annihilator graph under the extension of R to polynomial rings and rings of fractions. For instance, we show that the graphs AG(R) and AG(T(R)) are isomorphic, where T(R) is the total quotient ring of R. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo n, where n ⩾ 1.

More on the annihilator graph of a commutative ring

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 $Z(R)^*=Z(R)\setminus\{0\}$, 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, we study the affinity between annihilator graph and zero-divisor graph associated with a commutative ring. For instance, for a non-reduced ring $R$, it is proved that the annihilator graph and the zero-divisor graph of $R$ are identical to the join of a complete graph and a null graph if and only if $ann_R(Z(R))$ is a prime ideal if and only if $R$ has at most two associated primes. Among other results, under some assumptions, we give necessary and sufficient conditions under which $AG(R)$ is a star graph.

Annihilator graphs with four vertices

In this paper, we characterize all commutative semigroups with zero S such that their annihilator graphs have four vertices. Moreover, we give a complete (up to isomorphism) description of all pairs (G, S), where G is a graph and S is a commutative semigroup such that G is the annihilator graph of S and G has four vertices.

On the annihilator-ideal graph of commutative rings

Let R be a commutative ring with nonzero identity. We denote by AG(R) the annihilator graph of R, whose vertex set consists of the set of nonzero zero-divisors of R, and two distint vertices x and y are adjacent if and only if \(\mathrm {ann}(x) \cup \mathrm {ann}(y) \ne \mathrm {ann}(xy)\), where for \(t\in R\), we set \(ann(t) := \lbrace r\in R\ \vert \ rt=0\rbrace \). In this paper, we define the annihiator-ideal graph of R, which is denoted by \(A_{I}(R)\), as an undirected graph with vertex set \(A^*(R)\), and two distinct vertices I and J are adjacent if and only if \(\mathrm {ann}(I) \cup \mathrm {ann}(J) \ne \mathrm {ann}(IJ)\). We study some basic properties of \(A_{I}(R)\) such as connectivity, diameter and girth. Also we investigate the situations under which the graphs AG(R) and \(A_{I}(R)\) are coincide. Moreover, we examin the planarity of the graph \(A_{I}(R)\).

Coloring of the annihilator graph of a commutative ring

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 AG[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 ann[Formula: see text]. 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.

Final state interaction effect in the pure annihilation B→ϕϕ decay

We analyzed the process of [Formula: see text] decay in quantum chromodynamics factorization (QCDF) and final state interaction (FSI). In QCDF for this decay we have only the annihilation graph and we expected small branching ratio. Then we considered FSI effect as a sizable correction where the intermediate states are [Formula: see text], K+ K–, and [Formula: see text] mesons. To consider the amplitudes of these intermediate states, the QCDF approach was used. The experimental branching ratio of [Formula: see text] is less than 2 × 10–7 and our results are 0.04 × 10–7 and 1.54 × 10–7 from QCDF and FSI, respectively.

Final state interaction effect in pure annihilation Bs → ρρ decay

We analyzed the process of decay in quantum chromodynamics factorization (QCDF) and final state interaction (FSI) effects. In QCDF, for this decay, we have only the annihilation graph and we expected a small branching ratio. Then we considered the FSI effect as a sizable correction where the intermediate states are , , , and mesons. To consider the amplitudes of these intermediate states, the QCDF approach was used. The experimental branching ratio of is less than and our result is and from QCDF and FSI, respectively.

Final state interaction effects in B+→Ds*+ ϕ decay

In this article the exclusive decay of [Formula: see text] is calculated using the QCD factorization (QCDF) method and final state interaction (FSI). First, the [Formula: see text] decay is calculated via the QCDF method and only the annihilation graphs exist in that method. The result found using the QCDF method is lower than the experimental result. FSI is considered to solve the [Formula: see text] decay. For this decay, D+K0, D0K+, and [Formula: see text] via the exchange of K0, K+, and [Formula: see text] are chosen for the intermediate states and we calculate B+ → D+K0 → [Formula: see text] decay. The amplitude of B+ → D+K0 decay is calculated using the QCDF method again. The experimental branching ratio of [Formula: see text] decay is less than 1.2 × 10−5 and our results calculated using the QCDF method and FSI are (0.4 ± 0.06) × 10−7 and (0.93 ± 0.08) × 10−5, respectively.

The annihilator graph of a commutative semigroup

In this paper, we define and study the annihilator graph associated to a commutative semigroup with zero. Also, we characterize all annihilator graphs with three vertices.

Effects of final-state interactions in pure annihilation decay of B s 0 → π+π −

The hadronic decay of Bs0 → π+π− is analyzed by using “QCD factorization” (QCDF) method and final-state interaction (FSI). First, the Bs0 → π+π− decay is calculated via QCDF method and the annihilation graphs only exist in this method. Hence, the FSI must be seriously considered to solve the Bs0 → π+π− decay and the K+(*)K−(*) and \(K^{0(*)} K^{\bar 0(*)}\) via the exchange of K0(*) and K−(*) mesons are chosen for the intermediate states. To estimate the intermediate state amplitudes, the QCDF method is again used. These amplitudes are used in the absorptive part of the diagrams. The experimental branching ratio of Bs0 → π+π− decay is less than 1.2 × 10−6 and our results according to the QCDF method and FSI are 0.68 × 10−8 and 1.18 × 10−6, respectively.

Effects of final state interactions in $$B^ + \to D_S^ + \bar K^0$$ decay

We investigate the effects of final state interactions (FSI) contributions in the nonleptonic two body \(B^ + \to D_S^ + \bar K^0\) decay, however the hadronic decay of \(B^ + \to D_S^ + \bar K^0\) is analyzed by using “QCD factorization” (QCDF) method and final state interaction (FSI). First, the \(B^ + \to D_S^ + \bar K^0\) decay is calculated via QCDF method and only the annihilation graphs exist in that method. Hence, the FSI must be seriously considered to solve the \(B^ + \to D_S^ + \bar K^0\) decay and the D0π+(D0*ρ+), D+π0(D+*ρ0) and D+ηc(D+*J/ψ) via the exchange of K+(*), K0(*) and Ds+(*) mesones are chosen for the intermediate states. To estimate the intermediate states amplitudes, the QCDF method is again used. These amplitudes are used in the absorptive part of the diagrams. The experimental branching ratio of \(B^ + \to D_S^ + \bar K^0\) decay is less than 8 × 10−4 and the predicted branching ratio is 0.23 × 10−9 in the absence of FSI effects and it becomes 6.74 × 10−4 when FSI contributions are taken into account.

Some Properties of Annihilator Graph of a Commutative Ring

Let R be a commutative ring with unity. Let Z(R) be the set of all zero-divisors of R. For x  Z(R), let