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.

Nowadays, whereas the use of social networks and computer networks is increasing, the amount of associated complex data with Graph structure and their applications, such as classification, clustering, link prediction, and recommender systems, has risen significantly. Because of security problems and societal concerns, anomaly detection is becoming a vital problem in most fields. Applications that use a heterogeneous Graph, are confronted with many issues, such as different kinds of neighbors, different feature types, and differences in type and number of links. So, in this research, we employ the HetGNN model with some changes in loss functions and parameters for heterogeneous Graph embedding to capture the whole Graph features (structure and content) for anomaly detection, then pass it to a VAE to discover anomalous nodes based on reconstruction error. Our experiments on AMiner data set with many base-lines illustrate that our model outperforms state-of-the-arts methods in heterogeneous Graphs while considering all types of attributes.

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.