Let G be a (p; q) graph. Let f: V (G)! f1; 2; : : :; kg be a map where k 2 N and k > 1. For each edge uv, assign the label gcd(f(u); f(v)). f is called k-Total prime cordial labeling of G if jtf (i) 􀀀 tf (j)j  1, i; j 2 f1; 2;


; kg where tf (x) denotes the total number of vertices and the edges labelled with x. A graph with a k-total prime cordial labeling is called k-total prime cordial graph. In this paper we investigate the 4-total prime cordial labeling of some graphs like Prism, Helm, Dumbbell graph, Sun ower graph.

‎We obtain the exact relations of the Hosoya index that is defined as the sum of the number of all the matching sets‎, ‎on some classes of cycle-related graphs‎. ‎Moreover‎, ‎this index of three graph families‎, ‎namely‎, ‎chain triangular cactus‎, ‎Dutch windmill graph‎, ‎and Barbell graph is determined‎.

For any non-trivial abelian group A under addition a graph G is said to be A-magic if there exists a labeling f: E(G) → A − {0} such that, the vertex labeling f+ defined as f+(v) = P f(uv) taken over all edges uv incident at v is a constant. An A-magic graph G is said to be Zk-magic graph if the group A is Zk the group of integers modulo k. These Zk-magic graphs are referred to as k-magic graphs. In this paper we prove that the total graph, flower graph, generalized prism graph, closed Helm graph, lotus inside a circle graph, G ⊙ Km, m-splitting graph of a path and m-shadow graph of a path are Zk-magic graphs.

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.