The COMAXIMAL intersection GRAPH $CI(R)$ of ideals of a ring $R$ is an undirected GRAPH whose vertex set is the collection of all non-trivial (left) ideals of $R$ and any two vertices $I$ and  $J$ are adjacent if and only if $I+J=R$ and $I\cap J\neq0$. We study the connectedness of $CI(R)$. We also discuss  independence number, clique number, domination number, chromatic number of $CI(R)$.

In this paper, R is a commutative ring with a non-zero identity andM is a unital R-module. We introduce the COMAXIMAL colon ideal GRAPH C, (R) and colon submodule GRAPH C, (M),and study the interplay between the GRAPH-theoretic properties and the corresponding algebraic structures. C, (R) is a simple connected superGRAPH of the COMAXIMAL ideal GRAPH C(R) with diam(C, (R)) ,2. Moreover, we prove that if jV(C, (R)j ,3, then gr(C, (R)) = 3. We prove that if jMax(R)j = n, then C, (R) containing a complete n-partite subGRAPH. Also if M is a , nitely generated multiplication module, then C, (R) , = C, (M). Moreover, for Z-module Zn which n is not a prime, C, (Zn) , = Kd(n), where d(n) is the number of all divisors of the positive integer n other than 1 and n.

Let R be a commutative ring with identity. We use j (R) to denote the COMAXIMAL ideal GRAPH. The vertices of j (R) are proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I and J are adjacent if and only if I+J=R. In this paper we show some properties of this GRAPH together with planarity of line GRAPH associated to j (R).

The rings considered in this article are commutative with identity which admit at least two maximal ideals. This article is inspired by the work done on the COMAXIMAL ideal GRAPH of a commutative ring. Let R be a ring. We associate an undirected GRAPH to R denoted by G(R), whose vertex set is the set of all proper ideals I of R such that I ̸  J(R), where J(R) is the Jacobson radical of R and distinct vertices I1; I2 are adjacent in G(R) if and only if I1 \ I2 = I1I2. The aim of this article is to study the interplay between the GRAPH-theoretic properties of G(R) and the ring-theoretic properties of R.

Let R be a commutative ring with unity. The COMAXIMAL ideal GRAPH of R, denoted by C (R), is a GRAPH whose vertices are the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. In this paper, we classify all COMAXIMAL ideal GRAPHs with finite independence number and present a formula to calculate this number. Also, the domination number of C (R) for a ring R is determined. In the last section, we introduce all planar and toroidal COMAXIMAL ideal GRAPHs. Moreover, the commutative rings with isomorphic COMAXIMAL ideal GRAPHs are characterized. In particular we show that every finite COMAXIMAL ideal GRAPH is isomorphic to some C (Zn).

Let $G$ be a group and $S$ be the collection of all non-trivial proper subgroups of $G$. The co-maximal subgroup GRAPH $\Gamma(G)$ of a group $G$ is defined to be a GRAPH with $S$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. In this paper, we study the COMAXIMAL subgroup GRAPH on finite dihedral groups. In particular, we study order, maximum and minimum degree, diameter, girth, domination number, chromatic number and perfectness of COMAXIMAL subgroup GRAPH of dihedral groups. Moreover, we prove some isomorphism results on COMAXIMAL subgroup GRAPH of dihedral groups.

Let R be a commutative ring with identity 1≠0. The COMAXIMAL ideal GRAPH of R is the simple, undirected GRAPH whose vertex set is the set of all proper ideals of the ring R not contained in Jacobson radical of R and two vertices I and J are adjacent in this GRAPH if and only if I+J=R. In this article, we have discussed the GRAPH G(R) whose vertex set is the set of all proper ideals of ring R and two vertices I and J are adjacent in this GRAPH if and only if I+J≠R. In this article, we have discussed some interesting results about G(R) and its line GRAPH.

In this paper, we define the concept of pseudo-irreducible elements in multiplicative lattices and examine its relationship with other important concepts of multiplicative lattices such as prime elements, primary elements, and maximum elements, and then with the help of this concept, we define and analyze the complete COMAXIMAL factorization in multiplicative lattices. In particular, we characterize multiplicative lattices whose every element can be written as a product of pseudo-irreducible and pairwise COMAXIMAL elements.

The independence GRAPH Ind(G) of a GRAPH G is the GRAPH with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence GRAPH of Cartesian product of d copies of complete GRAPHs Kq, which is known as the Hamming GRAPH H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming GRAPH H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.