In this Ë‘work, we present theË‘ notion of the Ë‘graph for a KU-semigroup  as theË‘undirected simple graphË‘ with the vertices are the elementsË‘ of  and weË‘Ë‘study the Ë‘graph ofË‘ equivalence classesË‘ofË‘  which is determinedË‘ by theË‘ definition equivalenceË‘ relation ofË‘ these verticesË‘, andË‘ then some related Ë‘properties areË‘ given. Several examples are presented and some theorems are proved. ByË‘ usingË‘ the definitionË‘ ofË‘ isomorphicË‘ graph, Ë‘we showË‘ thatË‘ the graphË‘ of equivalence Ë‘classes Ë‘and the Ë‘graphË‘of Ë‘a KU-semigroup Ë‘  areË‘ theË‘ sameË‘,

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 .

     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

In this paper the concepts of weakly (resp., closure, strongly) Perfect Mappings are defined and the important relationships are studied: (a) Comparison between deferent forms of perfect mappings. (b) Relationship between compositions of deferent forms of perfect mappings. (c) Investigate relationships between deferent forms of perfect mappings and their graphs mappings.

This paper presents a grey model GM(1,1) of the first rank and a variable one and is the basis of the grey system theory , This research dealt  properties of grey model and a set of methods to estimate parameters of the grey model GM(1,1)  is the least square Method (LS) , weighted least square method (WLS), total least square method (TLS) and gradient descent method  (DS). These methods were compared based on two types of standards: Mean square error (MSE), mean absolute percentage error (MAPE), and after comparison using simulation the best method was applied to real data represented by the rate of consumption of the two types of oils a Heavy fuel (HFO) and diesel fuel (D.O) and has been applied several tests to

In this paper we show that if ? Xi is monotonically T2-space then each Xi is monotonically T2-space, too. Moreover, we show that if ? Xi is monotonically normal space then each Xi is monotonically normal space, too. Among these results we give a new proof to show that the monotonically T2-space property and monotonically normal space property are hereditary property and topologically property and give an example of T2-space but not monotonically T2-space.