‎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 Γ =(V, E) be a graph and W_( a)=\{w_1, … , w_k \} be a subset of the vertices of Γ and v be a vertex of it. The k-vector r_2 (v∣ W_a)=(a_Γ (v, w_1), … , a_Γ (v, w_k)) is the adjacency representation of v with respect to W in which a_Γ (v, w_i )=min\{2, d_Γ (v, w_i )\} and d_Γ (v, w_i ) is the distance between v and w_i in Γ . W_a is called as an adjacency resolving set for Γ if distinct vertices of Γ have distinct adjacency representations w. r. t W_a. The size of the smallest adjacency resolving set is the adjacency metric dimension of Γ and is denoted by dim_a (Γ ). In this paper, we prove that dim_a(Γ _E (Z_(P^n ) ))=⌈ (n-2)/2⌉ . Also, we show that Γ _E (Z_(p^2n ) )≅ Γ _E (R/I) in which p is a prime number, n is a natural number and I is a 2-absorbing ideal of the ring R which has a minimal primitive decomposition in the form of the intersection of n primitive ideals. Finally we conclude that dim_a⁡ 〖 (Γ _E (R/I))=n-1〗 .

Graph coloring is a crucial area of research in graph theory, with numerous algorithms proposed for various types of graph coloring, particularly graph p-distance coloring‎. In this study, we employ a recently introduced graph coloring algorithm to develop a hybrid algorithm approximating the chromatic number ‎p-distance, where $p$ represents a positive integer number. We apply our algorithm to molecular graphs as practical applications of our findings.

An optimal labeling of a graph with $n$ vertices and $m$ edges is an injective assignment of the first $n$ nonnegative integers to the vertices‎, ‎that induces‎, ‎for each edge‎, ‎a weight given by the sum of the labels of its end-vertices with the property that the set of all induced weights consists of the first $m$ positive integers‎. ‎We explore the connection of this labeling with other well-known functions such as super edge-magic and $\alpha$-labelings‎. ‎A graph with $n$ vertices is maximal when the number of edges is $2n-3$; all the results included in this work are about maximal graphs‎. ‎We determine the number of optimally labeled graphs using the adjacency matrix‎. ‎Several techniques to construct maximal graphs that admit an optimal labeling are introduced as well as a family of outerplanar graphs that can be labeled in this form.

‎Let $\mathcal{R}$ be the commutative ring $\mathcal{R}=\mathbb{Z}_{p^2}[x]/\langle x^{2} \rangle$ with identity and ${Z^{*}}(\mathcal{R})$ be the set of all non-zero zero-divisors of $\mathcal{R}$‎. ‎Then‎, ‎$\Gamma(\mathcal{R})$ is said to be a zero-divisor graph if and only if $a \cdot b= 0$ where $a,b \in V(\Gamma(\mathcal{R})) = {Z^{*}}(\mathcal{R})$ and $(a,b) \in E(\Gamma(\mathcal{R}))$‎. ‎Let $\lambda_1,\lambda_2,\dots,\lambda_n$ be the eigenvalues of the adjacency matrix‎, ‎and let $\mu_1,\mu_2,\dots,\mu_n$ be the eigenvalues of the Laplacian matrix of $\Gamma(\mathcal{R})$‎. ‎Then %the energy of $\Gamma(\mathcal{R})$ is defined as the sum of the absolute values of the eigenvalues of the graph $\Gamma(\mathcal{R})$ and the Laplacian energy of $\Gamma(\mathcal{R})$ is the sum of the absolute deviations of its Laplacian matrix's eigenvalues of the graph $\Gamma(\mathcal{R})$‎. ‎In this paper‎,‎we discuss the energy $\mathcal{E}(\Gamma(\mathcal{R}))=\sum_{i=1}^n \abs{\lambda_{i}}$ and the Laplacian energy $\mathcal{LE}(\Gamma(\mathcal{R}))=\sum_{i=1}^n \abs{\mu_{i}-\frac{2m}{n}}$ where $n$ and $m$ are the order and size of $\Gamma(\mathcal{R})$‎.

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Let be a simple and undirected graph with vertex set. A set is called a dominating set of if every vertex outside is adjacent to at least one vertex of. For any integer, a dominating set is called a-adjacency dominating set of if the induced subgraph contains at least one vertex of degree at most. The minimum cardinality of a-adjacency dominating set of is called the-adjacency domination number of that is denoted by. In this paper, the study of-adjacency domination in graphs is initiated, and exact values and some bounds on the-adjacency domination number of a given graph are presented. Furthermore, it is shown that there is a polynomial-time algorithm that computes the-adjacency domination number of a given tree. Moreover, it is proven that the decision problem associated to the-adjacency domination is NP-complete for bipartite graphs.