‎The adjacency matrix is important invariant of a graph with a chemical meaning‎, ‎when we study the chemical graphs‎. ‎In this paper‎, ‎the general form of the adjacency matrices of some hexagonal systems will be determined‎.

‎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 simple graph. The energy of $G$ isthe sum of absolute values of the eigenvalues of its adjacencymatrix $A(G)$.In this paper weconsider the edge energy of $G$ (or energy of line of $G$) which is definedas the absolute values of eigenvalues of edge adjacency matrix of $G$. We study the energy for the line of specific graphs.

In this paper the properties of node-node adjacency matrix in acyclic digraphs are considered. It is shown that topological ordering and node-node adjacency matrix are closely related. In fact, first the one to one correspondence between upper triangularity of node-node adjacency matrix and existence of directed cycles in digraphs is proved and then with this correspondence other properties of adjacency matrix in acyclic digraphs are presented.

Feature selection is crucial to improve the quality of classification and clustering. It aims to enhance machine learning performance and reduce computational costs by eliminating irrelevant or redundant features. However, existing methods often overlook intricate feature relationships and select redundant features. Additionally, dependencies are often hidden or inadequately identified. That’s mainly because of nonlinear relationships being used in traditional algorithms. To address these limitations, novel feature selection algorithms are needed to consider intricate feature relationships and capture high-order dependencies, improving the accuracy and efficiency of data analysis.In this paper, we introduce an innovative feature selection algorithm based on adjacency matrix, which is applicable to supervised data. The algorithm comprises three steps for identifying pertinent features. In the first step, the correlation between each feature and its corresponding class is measured to eliminate irrelevant features. Moving to the second step, the algorithm focuses on the selected features, calculates pairwise relationships and constructs an adjacency matrix. Finally, the third step employs clustering techniques to classify the adjacency matrix into k clusters, where k represents the number of desired features. From each cluster, the algorithm selects the most representative feature for subsequent analysis.This feature selection algorithm provides a systematic approach to identify relevant features in supervised data, thereby significantly enhance the efficiency and accuracy of data analysis. By taking into account both the linear and nonlinear dependencies between features and effectively detecting them across multiple feature sets, it successfully overcomes the limitations of previous methods.

Feature selection is crucial to improve the quality of classification and clustering. It aims to enhance machine learning performance and reduce computational costs by eliminating irrelevant or redundant features. However, existing methods often overlook intricate feature relationships and select redundant features. Additionally, dependencies are often hidden or inadequately identified. That’s mainly because of nonlinear relationships being used in traditional algorithms. To address these limitations, novel feature selection algorithms are needed to consider intricate feature relationships and capture high-order dependencies, improving the accuracy and efficiency of data analysis.In this paper, we introduce an innovative feature selection algorithm based on adjacency matrix, which is applicable to supervised data. The algorithm comprises three steps for identifying pertinent features. In the first step, the correlation between each feature and its corresponding class is measured to eliminate irrelevant features. Moving to the second step, the algorithm focuses on the selected features, calculates pairwise relationships and constructs an adjacency matrix. Finally, the third step employs clustering techniques to classify the adjacency matrix into k clusters, where k represents the number of desired features. From each cluster, the algorithm selects the most representative feature for subsequent analysis.This feature selection algorithm provides a systematic approach to identify relevant features in supervised data, thereby significantly enhance the efficiency and accuracy of data analysis. By taking into account both the linear and nonlinear dependencies between features and effectively detecting them across multiple feature sets, it successfully overcomes the limitations of previous methods.

This article explores the adjacency matrix of a two-sided group graph andits properties. We introduce the two-sided color group digraph to generalize the Cayley color graph and the two-sided group digraph. We alsoobtain the adjacency matrix of the latter digraph and provide a criterion fordetermining the normality of the adjacency matrix of a two-sided group graph.Moreover, we prove that if all the two-sided group digraphs of valency two fora certain group G are normal, then G is a Hamiltonian group. We also showthat if a strongly connected two-sided group digraph of valency two is normal,the corresponding group is isomorphic to the product of two groups: a cyclicgroup with either Tk,n or Hp,q, or an abelian group.

In this paper, we present an edge detection method based on wavelet transform and Hessian matrix of image at each pixel. Many methods which based on wavelet transform, use wavelet transform to approximate the gradient of image and detect edges by searching the modulus maximum of gradient vectors. In our scheme, we use wavelet transform to approximate Hessian matrix of image at each pixel, too. The main idea of our methods lies in the fact that, the direction of largest surface curvature is the eigenvector of the Hessian matrix corresponding to the largest absolute eigenvalue. Infact, we use the Hessian matrix's information to increase or decrease the effect of wavelet transform in and directions.