Let  be a non-trivial simple graph. A dominating set in a graph is a set of vertices such that every vertex not in the set is adjacent to at least one vertex in the set. A subset  is a minimum neighborhood dominating set if  is a dominating set and if for every  holds. The minimum cardinality of the minimum neighborhood dominating set of a graph  is called as minimum neighborhood dominating number and it is denoted by  . A minimum neighborhood dominating set is a dominating set where the intersection of the neighborhoods of all vertices in the set is as small as possible, (i.e., ). The minimum neighborhood dominating number, denoted by , is the minimum cardinality of a minimum neighborhood dominating set. In other words, it is the

Let A ⊆ V(H) of any graph H, every node w of H be labeled using a set of numbers; , where d(w,v) denotes the distance between node w and the node v in H, known as its open A-distance pattern. A graph H is known as the open distance-pattern uniform (odpu)-graph, if there is a nonempty subset A ⊆V(H) together with  is the same for all . Here  is known as the open distance pattern uniform (odpu-) labeling of the graph H and A is known as an odpu-set of H. The minimum cardinality of vertices in any odpu-set of H, if it exists, will be known as the odpu-number of the graph H. This article gives a characterization of maximal outerplanar-odpu graphs. Also, it establishes that the possible odpu-number of an odpu-maximal outerplanar graph i

F index is a connected graph, sum of the cubes of the vertex degrees. The forgotten topological index has been designed to be employed in the examination of drug molecular structures, which is extremely useful for pharmaceutical and medical experts in understanding the biological activities. Among all the topological indices, the forgotten index is based on degree connectivity on bonds. This paper characterized the forgotten index of union of graphs, join graphs, limits on trees and its complements, and accuracy is measured. Co-index values are analyzed for the various molecular structure of chemical compounds

The main purpose of this paper, is to introduce a topological space , which is induced by reflexive graph and tolerance graph , such that  may be infinite. Furthermore, we offered some properties of  such as connectedness, compactness, Lindelöf and separate properties. We also study the concept of approximation spaces and get the sufficient and necessary condition that topological space is approximation spaces.

This paper is devoted to the discussion the relationships of connectedness between some types of graphs (resp. digraph) and Gm-closure spaces by using graph closure operators.

     A graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense related. The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). A directed graph is a graph in which edges have orientation. A simple graph is a graph that does not have more than one edge between any two vertices and no edge starts and ends at the same vertex.  For a simple undirected graph G with order n, and let  denotes its complement. Let δ(G), ∆(G) denotes the minimum degree and maximum degree of G respectively. The complement degree polynomial of G is the polynomial CD[G,x]= , where C

Antimagic labeling of a graph  with  vertices and  edges is assigned the labels for its edges by some integers from the set , such that no two edges received the same label, and the weights of vertices of a graph  are pairwise distinct. Where the vertex-weights of a vertex  under this labeling is the sum of labels of all edges incident to this vertex, in this paper, we deal with the problem of finding vertex antimagic edge labeling for some special families of graphs called strong face graphs. We prove that vertex antimagic, edge labeling for strong face ladder graph , strong face wheel graph ,  strong face fan graph , strong face prism graph  and finally strong face friendship graph .

The metric dimension and dominating set are the concept of graph theory that can be developed in terms of the concept and its application in graph operations. One of some concepts in graph theory that combine these two concepts is resolving dominating number. In this paper, the definition of resolving dominating number is presented again as the term dominant metric dimension. The aims of this paper are to find the dominant metric dimension of some special graphs and corona product graphs of the connected graphs  and , for some special graphs  . The dominant metric dimension of  is denoted by  and the dominant metric dimension of corona product graph G and H is denoted by .