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Let R be a commutative ring with unity 1 ≠ 0 and let R× be the set of all unit elements of R. The unitary Cayley graph of R, denoted by GR = Cay(R, R×), is a simple graph whose vertex set is R and there is an edge between two distinct vertices x and y of R if and only if x − y ∈ R×. In this paper, we determine the Laplacian and signless Laplacian eigenvalues for the unitary Cayley graph of a commutative ring. Also, we compute the Laplacian and signless Laplacian energy of the graph GR and its line graph.

Let (P, ≤) be a poset with the least element 0. The intersection graph of ideals of P, denoted by G(P), is a graph whose vertices are all nontrivial ideals of P and two distinct vertices I and J are adjacent if and only if I ∩ J ≠ {0}. In this paper, we study the planarity and outerplanarity of the intersection graph G(P). Also, we determine all posets with split intersection graphs.

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]

Let R be a commutative ring without identity. The zero-divisor graph of R; denoted by -(R) is a graph with vertex set Z(R)nf0g which is the set of all nonzero zero-divisor elements of R; and two distinct vertices x and y are adjacent if and only if xy = 0: In this paper, we characterize the rings whose zero-divisor graphs are ring graphs and outerplanar graphs. Further, we establish the planar index, ring index and outerplanar index of the zero-divisor graphs of finite commutative rings without identity. [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]

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.

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 L be a lattice, 𝒥(L) be the set of ideals of L and S be a subset of 𝒥 (L). In this paper, we introduce an undirected Cayley graph of L, denoted by ΓL,S with elements of 𝒥 (L) as the vertex set and, for two distinct vertices I and J, I is adjacent to J if and only if there is an element K of S such that I ∨ K = J or J ∨ K = I. We study some basic properties of the graph ΓL,S such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of ΓL,S.

In this paper, we study the planar and outerplanar indices of some graphs associated to a commutative ring. We give a full characterization of these graphs with respect to their planar and outerplanar indices when R is a finite ring.

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]