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Let R be a commutative ring with identity. The nilpotent graph of R, denoted by ΓN(R), is a graph with vertex set ZN(R)∗, and two vertices x and y are adjacent if and only if xy is nilpotent, where ZN(R) = {x ∈ R∣xy is nilpotent, for some y ∈ R∗}. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. In this paper, we characterize all rings whose ΓN(R) is perfect. In addition, it is shown that for a ring R, if R is Artinian, then ω(ΓN(R)) = χ(ΓN(R)) = |Nil(R)∗| + |Max(R)|.

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R is the undirected (simple) graph WΓ(R) with vertex set Z(R) ∗, and two distinct vertices x and y are adjacent if and only if there exist r ∈ ann(x) and s ∈ ann(y) such that rs = 0. It follows that WΓ(R) contains the zero-divisor graph Γ(R) as a subgraph. In this paper, the connectedness, diameter, and girth of WΓ(R) are investigated. Moreover, we determine all rings whose weakly zero-divisor graphs are star. We also give conditions under which weakly zero-divisor and zero-divisor graphs are identical. Finally, the chromatic number of WΓ(R) is studied.

Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by AR, is a graph whose vertex set is the set of all non-zero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann(I) ∩ Ann(J) = (0). In this paper, we characterize all Artinian rings for which both of the graphs AR and AR (the complement of AR), are chordal. Moreover, all Artinian rings whose AR (and thus AR) is perfect are characterized.

Let R be a commutative ring with identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R and two distinct vertices x and y are adjacent if and only if $$x+y$$ is a unit of R. In this paper, perfect unit graphs of Artinian rings are investigated.

Let $R$ be a commutative ring with unity. The annihilator ideal graph of $R$, denoted by $\Gamma _{\mathrm{Ann}} (R) $, is a graph whose vertices are all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if$ I \cap \mathrm{Ann} _{R} (J) \neq \lbrace 0\rbrace $ or $J \cap \mathrm{Ann} _{R} (I) \neq \lbrace 0\rbrace $.In this paper, all rings with planar annihilator ideal graphs are classified.Furthermore, we show that all annihilator ideal graphs are perfect. Among other results, it is proved that if $\Gamma _{\mathrm{Ann}} (R) $ is a tree, then $\Gamma _{\mathrm{Ann}} (R) $ is star.

In this paper, we introduce the notion of classical 2-prime subsemimodules of a semimodule M over a commutative semiring S with identity, which is a generalization of classical prime subsemimodules. A proper subsemimodule N of M having the property that for each [Formula: see text] and each subsemimodule K of M, the inclusion [Formula: see text] implies [Formula: see text] or [Formula: see text] is called classical 2-prime. We give some results concerning classical 2-prime subsemimodules. Also, the classical 2-prime avoidance theorem for subsemimodules is proved. Finally, the notion of valuation semimodules is introduced and some results on classical 2-prime subsemimodules in valuation semimodules are given.

Let R be a commutative ring with non-zero identity and $$W^*(R)$$ be the set of all non-zero and non-unit elements of R. The cozero-divisor graph of R, denoted by $$\Gamma ^{\prime }(R)$$ , is a graph with the vertex set $$W^*(R)$$ and two distinct vertices a and b are adjacent if and only if $$a\not \in Rb$$ and $$b\not \in Ra$$ . In this paper, the metric dimension and strong metric dimension of $$\Gamma ^{\prime }(R)$$ are investigated. We compute the exact values of strong metric and metric dimension in cozero-divisor graphs of reduced rings. Moreover, the metric dimension in cozero-divisor graphs of non-reduced rings is discussed.

Let [Formula: see text] be a commutative ring with identity. In this paper, we introduce the concept of [Formula: see text]-absorbing prime ideals which is a generalization of prime ideals. A proper ideal [Formula: see text] of [Formula: see text] is called [Formula: see text]-absorbing prime if for all nonunit elements [Formula: see text] such that [Formula: see text], then either [Formula: see text] or [Formula: see text]. Some properties of [Formula: see text]-absorbing prime are studied. For instance, it is shown that if [Formula: see text] admits a [Formula: see text]-absorbing prime ideal that is not a prime ideal, then [Formula: see text] is a quasi–local ring. Among other things, it is proved that a proper ideal [Formula: see text] of [Formula: see text] is [Formula: see text]-absorbing prime if and only if the inclusion [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text] implies that [Formula: see text] or [Formula: see text]. Also, [Formula: see text]-absorbing prime ideals of PIDs, valuation domains, Prufer domains and idealization of a modules are characterized. Finally, an analogous to the Prime Avoidance Theorem and some applications of this theorem are given.

In this paper, five degree based topological indices, the first Zagreb (M 1), second Zagreb (M 2), first multiple Zagreb (PM 1), second multiple Zagreb (PM 2), and the hyper Zagreb (HM) indices of two types of dendrimers are studied. In addition, two distance based topological indices, the total eccentricity (θ) and eccentric connectivity (ξc ) indices of these dendrimers are computed.

Let R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

Let α be an endomorphism of an arbitrary ring $$R$$ with identity. The aim of this article is to introduce the ideal $$I$$ of $$R$$ as an α-skew M-McCoy which is a generalization of α-rigid ideals and McCoy ideals and to investigate its properties. α-skew M-McCoy is an α-skew McCoy ideals relative to a monoid $$M$$. The findings show that if $${\alpha ^t} = {I_R}$$ for some positive integer $$t$$ and $$I$$ be an α-skew M-McCoy left ideal of $$R$$ then $$I[M,\alpha ]$$ is α-skew M-McCoy left ideal of $$R[M,\alpha ]$$. Also if $$R$$ be a locally finite ring and $$I$$ be an α-skew M-McCoy left ideal of $$R$$, then $$I$$ is α-semicommutative left ideal of $$R$$.

Let $R$ be a commutative ring with non-zero identity. The cozero-divisor graph of $R$, denoted by $\Gamma^{\prime}(R)$, is a graph with vertices in $W^*(R)$, which is the set of all non-zero and non-unit elements of $R$, and two distinct vertices $a$ and $b$ in $W^*(R)$ are adjacent if and only if $a\not\in Rb$ and $b\not\in Ra$. In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given., Comment: 9 pages

The concept of nil-M-McCoy (nil-McCoy ring relative to monoid M), which are generalizations of McCoy ring and nil-M-Armendariz rings have been introduced, and we investigate their properties. It is shown that every NI ring is nil-M-McCoy for any unique product monoid M, it has also been shown that every semicommutative rings is nil-M-McCoy for any unique product monoid and any strictly totally ordered monoid M. Moreover, it is proved that for an ideal I of R, if I is semicommutative and R / I is nil-M-McCoy then R is nil-M-McCoy for any strictly totally ordered monoid. We extend and unify many known results related to McCoy rings and nil-Armendariz ring.

For a monoid M, we introduce strongly M-semicommutative rings obtained as a generalization of strongly semicommutative rings and investigate their properties. We show that if G is a finitely generated Abelian group, then G is torsion free if and only if there exists a ring R with |R| ≥ 2 such that R is strongly G-semicommutative.

In this paper some basic mathematical properties for the third and hyper Zagreb coindices of graph operations containing the Cartesian product and composition will be explained.

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.

We generalize the concepts of semicommutative, skew Armendariz, Abelian, reduced, and symmetric left ideals and study the relationships between these concepts.

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The extended zero-divisor graph of $R$ is the undirected (simple) graph $\Gamma'(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if either $Rx\cap \mathrm{Ann}(y)\neq (0)$ or $Ry\cap \mathrm{Ann}(x)\neq (0)$. It follows that the zero-divisor graph $\Gamma(R)$ is a subgraph of $\Gamma'(R)$. It is proved that $\Gamma'(R)$ is connected with diameter at most two and with girth at most four, if $\Gamma'(R)$ contains a cycle. Moreover, we characterize all rings whose extended zero-divisor graphs are complete or star. Furthermore, we study the affinity between extended zero-divisor graph and zero-divisor graph associated with a commutative ring. For instance, for a non-reduced ring $R$, it is proved that the extended zero-divisor 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.

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.

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, all rings whose annihilator graphs can be embed on the plane or torus are classified., 11 pages, 1 figure

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

A recently published paper [6] considered the total graph of commutative ring R. In this paper, we compute Wiener, hyper-Wiener, reverse Wiener, Randić, Zagreb, ABC and GA indices of zero-divisor graph.

Let ? be an endomorphism of a ring R. We introduce the notion of weak ?-skew McCoy rings which are a generalization of the ?-skew McCoy rings and the weak McCo rings. Some properties of this generalization are established, and connections of properties of a weak ?-skew McCoy ring R with n ? n upper triangular Tn(R) are investigated. We study relationship between the weak skew McCoy property of a ring R and its polynomial ring, R[x]. Among applications, we show a number of interesting properties of a weak ?-skew McCoy ring R such as weak skew McCoy property in a ring R.

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph 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 affinity between the annihilator-ideal graph and the annihilating-ideal graph [Formula: see text] (a well known graph with the same vertices and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]) associated with [Formula: see text]. All rings whose [Formula: see text] and [Formula: see text] are characterized. Among other results, we obtain necessary and sufficient conditions under which [Formula: see text] is a star graph.

Let G be a graph. The core of G, denoted by G Δ, is the subgraph of G induced by the vertices of degree Δ(G), where Δ(G) denotes the maximum degree of G. A k -edge coloring of G is a function f : E(G) → L such that |L| = k and f (e 1) ≠ f (e 2) for all two adjacent edges e 1 and e 2 of G. The chromatic index of G, denoted by χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is said to be Class 1 if χ′(G) = Δ(G) and Class 2 if χ′(G) = Δ(G) + 1. In this paper it is shown that every connected graph G of even order whose core is a cycle of order at most 13 is Class 1.

Let R be a ring with non-zero identity and U(R) be the group of units of R. The unit graph of R, denoted by G(R), is a graph defined on the elements of R, and two distinct vertices r and s are adjacent if and only if r+s∈U(R). We investigate connectivity, diameter and the girth of the unit graph of a left Artinian ring. Also, by providing an algorithm, we determine when the unit graph of a finite ring is Hamiltonian.

Let R be a commutative ring and Max (R )b e the set of max- imal ideals of R. The regular digraph of ideals of R, denoted by − − → Γreg(R), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R ,d e- noted by Γreg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian ring R ,w e prove that|Max (R) |− 1 ω(Γreg(R)) |Max (R)| and χ(Γreg(R)) =2 |Max (R) |− k − 1, where k is the number of fields, appeared in the decompo- sition of R to local rings. Among other results, we prove that − − → Γreg(R) is strongly connected if and only if R is an integral domain. Finally, the diameter and the girth of the regular graph of ideals of Artinian rings are determined.

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ T 1 (N, M) = 0 (resp. Γ 1 T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I n (T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.

In this paper, we introduce a class of R- modules called hw-divisible modules and then we investigate some properties of them.

Let [Formula: see text] be a commutative ring with identity which is not an integral domain. An ideal [Formula: see text] of a ring [Formula: see text] is called an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. Let [Formula: see text] be a simple undirect graph associated with [Formula: see text] whose vertex set is the set of all nonzero annihilating ideals of [Formula: see text] and two distinct vertices [Formula: see text] are joined if and only if [Formula: see text] is also an annihilating ideal of [Formula: see text]. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. In this paper, we characterize all rings whose [Formula: see text] is perfect.

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 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 for which the genus of [Formula: see text] is two.

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals with nonzero annihilator. The strongly annihilating-ideal 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] and [Formula: see text]. It is proved that [Formula: see text] is connected with diameter at most two and with girth at most four, if it contains a cycle. Moreover, we characterize all rings whose strongly annihilating-ideal graphs are complete or star. Furthermore, we study the affinity between strongly annihilating-ideal graph and annihilating-ideal graph (a well-known graph with the same vertices and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]) associated with a commutative ring.

Let R be a ring with unity. The nilpotent graph of R, denoted by GN(R), is a graph with vertex set ZN(R)* = {0 \neq x \in R \mid xy \in N(R) for some 0 \neq y \in R}; and two distinct vertices x and y are adjacent if and only if xy \in N(R), where N(R) is the set of all nilpotent elements of R. Recently, it has been proved that if R is a left Artinian ring, then diam(GN(R)) \leq 3. In this paper, we present a new proof for the above result, where R is a finite ring. We study the diameter and the girth of matrix algebras. We prove that if F is a field and n \geq 3, then diam(GN(Mn(F))) = 2. Also, we determine diam (GN (M2(F))) and classify all finite rings whose nilpotent graphs have diameter at most 3. Finally, we determine the girth of the nilpotent graph of matrix algebras.

Let R be a commutative ring with identity and A(R) be the set of ideals with non-zero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)∗ = A(R)\{0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study some connections between the graph theoretic properties of this graph and some algebraic properties of a commutative ring. We prove that if AG(R) is a tree, then either AG(R) is a star graph or a path of order 4 and in the latter case R ∼= F × S, where F is a field and S is a ring with a unique non-trivial ideal. Moreover; we prove that if R has at least three minimal prime ideals, then AG(R) is not a tree. It is shown that for every reduced ring R, if R has at least three minimal prime ideals, then AG(R) contains a triangle. Also we prove that for every non-reduced ring R, if |Min(R)| = 2, then either AG(R) contains a cycle or AG(R) ∼= P4. Finally, it is proved that, if |Min(R)| = 1 and AG(R) is a bipartite graph, then AG(R) is a star graph.

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.

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 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 ann[Formula: see text] is an essential ideal. It is proved that [Formula: see text] is connected with diameter at most three and with girth at most four, if [Formula: see text] 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.

Let $G$ be a graph. The core of $G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by the vertices of degree $\Delta(G)$, where $\Delta(G)$ denotes the maximum degree of $G$. A $k$-edge coloring of $G$ is a function $f:E(G)\rightarrow L$ such that $|L| = k$ and $f(e_1)\neq f(e_2)$ for all two adjacent edges $e_1$ and $e_2$ of $G$. The chromatic index of $G$, denoted by $\chi'(G)$, is the minimum number $k$ for which $G$ has a $k$-edge coloring. A graph $G$ is said to be Class $1$ if $\chi'(G) = \Delta(G)$ and Class $2$ if $\chi'(G) = \Delta(G) + 1$. In this paper it is shown that every connected graph $G$ of even order and with $\Delta(G_{\Delta})\leq 2$ is Class $1$ if $|G_{\Delta}|\leq 9$ or $G_{\Delta}$ is a cycle of order $10$.

We introduce the concept of weak α-skew Armendariz ideals and investigate their properties. Moreover, we prove that I is a weak α-skew Armendariz ideal if and only if I[x] is a weak α-skew Armendariz ideal. As a consequence, we show that R is a weak α-skew Armendariz ring if and only if R[x] is a weak α-skew Armendariz ring. Введено поняття слабких α-косих iдеалiв Армендарiза та дослiджено їх властивостi. Крiм того, доведено, що I є слабким α-косим iдеалом Армендарiза тодi i тiльки тодi, коли I[x] є слабким α-косим iдеалом Армендарiза. Як наслiдок, показано, що R є слабким α-косим кiльцем Армендарiза тодi i тiльки тодi, коли R[x] є слабким α-косим кiльцем Армендарiза.

In this research study, several topological indices have been investigated for linear [n]-Tetracene, V- Tetracenic nanotube, H-Tetracenic nanotube and Tetracenic nanotori. The calculated indices are first, second, third and modified second Zagreb indices. In addition, the first and second Zagreb coindices of these nanostructures were calculated. The explicit formulae for connectivity indices of various families of Tetracenic nanotubes and nanotori are presented in this manuscript. These formulae correlate the chemical structure of nanostructures to the information about their physical features. J.Natn.Sci.Foundation Sri Lanka 2015 43 (2): 127 - 133

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

Let R be a ring with unity and I(R)* be the set of all nontrivial left ideals of R. The intersection graph of ideals of R, denoted by G(R), is a graph with the vertex set I(R)* and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ 0. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose intersection graphs of ideals are not connected. Also we determine all rings whose clique number of the intersection graphs of ideals is finite. Among other results, it is shown that for a ring R, if the clique number of G(R) is finite, then the chromatic number is finite and if R is a reduced ring, then both are equal.

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.

This paper has been withdrawn by the author because there are some typos in proofs., Comment: This paper has been withdrawn by the author because there are some typos in proofs