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Let L be a lattice. For a natural number n, we associate a simple graph, denoted by ΓnL, with Ln\ {0} as the vertex set and two distinct vertices X and Y being adjacent if and only if there exists an n×n lower triangular matrix A over L such that AXT = YT or AYT = XT, where for a matrix B, BT is the matrix transpose of B. In this paper, we study some basic properties of ΓnL. Also, we investigate the genus number of the generalized Cayley graph ΓnL. [ABSTRACT FROM AUTHOR]

In this paper, we study the planar and outerplanar indices of the comaximal and cozero-divisor graphs and give a full characterization of these graphs with respect to their planar and outerplanar indices. [ABSTRACT FROM AUTHOR]

Let (P,≤) be a partially ordered set with a least element 0. In this paper, we define and study the cozero-divisor graph of P, which is an (undirected) graph with vertex-set P\{0}, and two distinct vertices x and y are adjacent if and only if x ∊ {y}e and y ∊ {x}e, where for a vertex z, {z}e denotes the set of lower bounds of z. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity. In this paper we intro- duce the coannihilator graph of R, which is a dual of the annihilator graph AG(R), denoted by AG'(R). AG'(R) is a graph with the vertex set W*(R), where W*(R) is the set of all nonzero and nonunit elements of R, and two distinct vertices x and y are adjacent if and only if x ∉ xyR or y ∉ xyR, where for z ∊ R, zR is the principal ideal generated by z. We study the interplay between the ring-theoretic properties of R and graph-theoretic properties of AG'(R). Also we completely determine all finite commutative rings R such that AG'(R) is planar, outerplanar or ring graph. Among other things, we prove that AG'(R) has a cut vertex if and only if R is isomorphic to Z∊2 × K, where K is a field. Also, we examine the domination number of AG'(R). [ABSTRACT FROM AUTHOR]

Let R be a Noetherian commutative ring. The regular digraph of ideals of R introduced in [10] which denoted by .... The vertex set of ... is set of nontrivial ideals of R and I → J is an arc in Γ(R) if and only if I ⊈ Z(J). In this paper we will introduce a dual of ... and denote it by Γ*(R). The vertex set of Γ*(R) is set of nontrivial ideals of R and I → J is an arc in Γ*(R) if and only if there exists an element x in I such that the R-module homomorphism fx: J → J is an epimorphism by the rule fx(y) = yx. We investigate the interplay between Γ*(R) and ring-theoretic properties of R and graph-theoretic properties of Γ*(R). Also we study the relation between two graphs Γ(R) and Γ*(R). In particular we will show that the value of ω(Γreg(R)) in [10] is not correct, and we will compute the correct value of ω(Γreg(R)). [ABSTRACT FROM AUTHOR]

For a commutative ring R with non-zero identity, we denote the set of nonzero elements x in R with xR ≠ R by W*(R). So W*(R) is the set of non-units of R. In this paper, we introduce the cozero-divisor graph Γ'(R), which is a dual of zero-divisor graph Γ(R) 'in some sense', as the (undirected) graph with vertices W*(R) and for two distinct elements x and y in W*(R), the vertices x and y are adjacent if and only if x ∉ yR and y ∉ xR. We investigate the interplay between the ring-theoretic properties of R and graph-theoretic properties of Γ'(R). Also we study the relations between two graphs Γ(R) and Γ'(R). [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with non-zero identity. For a natural number n, we associate a directed graph, denoted byPnR,with Rn\ {0}as the vertex set and, for two distinct verticesX = (x1, x2, . . . , xn) and Y = (y1, y2, . . . , yn),we have an arc X → Y in PnRif and only if there exists an n × n lower triangular matrix A =(aij ) over R such that AXT = Y Tand det (A) ϵ [x1, x2, . . . , x2], where [x1, x2, . . . , xn] is a subsemigroup generated by the elements x1, x2, . . . , xn in the multiplicative semigroup R. When we consider the ring R as a semigroup with respect to multiplication, the graph P1R is the usual directed power graph (over a semigroup). Hence PnR is a gener- alization of power graphs. In this paper we study some basic properties of PnR. Also we study the planarity, outer planarity and ring graph of PnR and we determine the cases for which the graph PnR has thickness 2. [ABSTRACT FROM AUTHOR]