Let $G=(V,E)$ be a graph of order $n$ and size $m.$ The graph $Sp(G)$ obtained from $G$ by adding a new vertex $v'$ for every vertex $v\in V$ and joining $v'$ to all neighbors of $v$ in $G$ is called the splitting graph of $G.$ In this paper, we determine the domination number, the total domination number, connected domination number, paired domination number and independent domination number for the splitting graph $Sp(G).$

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.

In this paper, we propose some new preconditioners  for solving multilinear system $\mathcal{A}\mathbf{x}^{m-1}=\mathbf{b}$. These preconditioners are based on tensor splitting. We also present some theorems for analyzing and convergence of the preconditioned  Jacobi-, Gauss-Seidel-, and SOR-type iterative methods. Numerical examples are presented to verify the efficiency of the proposed preconditioned methods.

During the past years, one of the most important problems has been air pollution in urban areas. In this regard, ozone, as one of the major products of photochemical reactions, has great importance. The term "photochemical" is applied to a number of secohdary pollutants that appear as a result of sun-related reactions, ozone being the most important one. So far various models have been suggested to predict these pollutants. In this paper, we developed the model that has been introduced by Dabir, et al. [4]. In this model model than 48 chemical species and 114 chemical reactions are involved. The resluts of this development, showed good to excellent agreement across the region for coni pounds such as O3, NO, NO2, CO, and SO2 with regard to VOC and NMHC. The results of the simulation were compared with previous work [4] and the effects of increashlg the number of componets and reactions were evaluated. The results of the operator splitting method were compared witll non splitting solving method. The result showed that splitting method with one-tenth time step collapsed with non splitting method (Crank-Nicolson, under-relaxation.iteration method without splitting of the equation terms). Then we developed One dimensional model to 3-D and results were compared witll experimental data.

Let R be a commutative ring and M be an R-module. The M-intersection graph of ideals of R, denoted by GM(R) is a graph with the vertex set I(R) ∗, , and two distinct vertices I and J are adjacent if and only if IM ∩,JM ̸, = 0. In this paper, we study GR/J (R/I), where I and J are ideals of R and I ⊆,J. We characterize all ideals I and J for which GR/J (R/I) is planar, outerplanar or ring graph.