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.

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.

‎Graph coloring is the assignment of one color to each vertex of a Graph so that two adjacent vertices are not of the same color‎. ‎The Graph coloring problem (GCP) is a matter of combinatorial optimization‎, ‎and the goal of GCP is determining the chromatic number $\chi(G)$‎. ‎Since GCP is an NP-hard problem‎, ‎then in this paper‎, ‎we propose a new approximated algorithm for finding the coloring number (it is an approximation of chromatic number) by using a Graph adjacency matrix to colorize or separate a Graph‎. ‎To prove the correctness of the proposed algorithm‎, ‎we implement it in MATLAB software‎, ‎and for analysis in terms of solution and execution time‎, ‎we compare our algorithm with some of the best existing algorithms that are already implemented in MATLAB software‎, ‎and we present the results in tables of various Graphs‎. ‎Several available algorithms used the largest degree selection strategy‎, ‎while our proposed algorithm uses the Graph adjacency matrix to select the vertex that has the smallest degree for coloring‎. ‎We provide some examples to compare the performance of our algorithm to other available methods‎. ‎We make use of the Dolan-Mor\'e performance profiles to assess the performance of the numerical algorithms‎, ‎and demonstrate the efficiency of our proposed approach in comparison with some existing methods‎.

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 $Sp(G).$

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In this paper, exact formulas for the dependence, independence, vertex cover and clique polynomials of the power Graph and its superGraphs for certain finite groups are presented.

Let $G=(V,E)$ be a connected Graph with the vertex-set $V$ and  the edge-set $E$.    The subdivision Graph $S(G)$ of the Graph $G$ is obtained from $G$ by adding a vertex in the middle of every edge of $G$.  In this paper, we investigate some properties of the Graphs  $S(G)$ and $L(S(G))$, where $L(S(G))$ is the line Graph of $S(G)$. We will see that $S(G)$ and  $L(S(G))$  inherit some  properties of $G$.    For instance, we show that if $G \ncong C_n$, then $Aut(G) \cong Aut(L(S(G)))$ (as abstract groups), where $C_n$ is the cycle of order $n$.