‎Since the problem of computing the Adjacency dimension of a graph is NP-hard‎, ‎finding the Adjacency dimension of special classes of graphs or obtaining good bounds on this invariant is valuable‎. ‎In this paper we determine the properties of each Adjacency resolving set of paths. ‎Then, ‎by ‎using ‎these ‎properties,‎ we determine the Adjacency dimension of broom and double broom graphs‎.

‎For an ordered set $W=\{w_1,w_2,\ldots,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),\ldots,d(v,w_k))$ is  called  the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called  a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension. In this paper, we investigate the metric dimension of the lexicographic product  of graphs $G$ and $H$, $G[H]$, for some known graphs.

‎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.

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.

Let $G=(V(G),E(G))$ be a simple and connected graph. The distance between any two vertices $x$ and $y$, denoted by $d_G(x,y)$, is defined as the length of a shortest path connecting $x$ and $y$ in $G$.The degree of a vertex $x$ in $G$, denoted by $\deg_G(x)$, is defined as the number of vertices in $G$ of distance one from $x$.The eccentric Adjacency index (briefly EAI) of a connected graph $G$ is defined as\[\xi^{ad} (G)=\sum_{u\in V(G)}\se_G(u)\varepsilon_G(u)^{-1},\]\noindentwhere $\se_G(u)=\displaystyle\sum_{\substack{v\in V(G)\\ d_G(u,v)=1}}\deg_{G}(v)$ and$\varepsilon_G(u)=\max \{d_G(u,v)\mid v \in V(G)\}$.In this article, we aim to obtain all extremal graphs based on the value ofEAI among all simple and connected graphs, all trees, and all trees with perfect matching.