Let R be an associative ring. The essential purpose of the present paper is to introduce the concept of generalized commuting mapping of R. Let U be a non-empty subset of R, a mapping   : R  R is called a generalized commuting mapping on U if there exist a mapping :R R such that =0, holds for all U. Some results concerning the new concept are presented.

Background: Dental stone casts come into contact with impression materials and becomes susceptible to cross contamination from saliva and blood. This study was done to evaluate the physical and mechanical properties of dental stone type IV after treatments with various disinfecting agents and regimes (methods). Materials and Methods: Type IV dental stone and different types of disinfecting agents were used and divided into seven groups: G1: dental stone without disinfection (control group), G2: dental stone mixed with silver nitrate powder 0.5% , G3: dental stone mixed with silver nitrate powder 1%, G4: dental stone mixed with copper sulfate powder 0.5%, G5: dental stone mixed with copper sulfate powder 1% ,G6: dental stone immersed in prop

Consider the (p,q) simple connected graph . The sum absolute values of the spectrum of quotient matrix of a graph  make up the graph's quotient energy. The objective of this study is to examine the quotient energy of identity graphs and zero-divisor graphs  of commutative rings using group theory, graph theory, and applications. In this study, the identity graphs  derived from the group  and a few classes of zero-divisor graphs  of the commutative ring R are examined.

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 .

In this paper we introduce the definition of  Lie ideal on inverse semiring and we generalize some results of Herstein about Lie structure of an associative rings to inverse semirings.

In this paper, Nordhaus-Gaddum type relations on open support independence number of some derived graphs of path related graphs under addition and multiplication are studied.

The topological indices are functions on the graph that do not depend on the labeling of their vertices. They are used by chemists for studying the properties of chemical compounds.  Let  be a simple connected graph. The Hyper-Zagreb index of the graph ,  is defined as  ,where  and  are the degrees of vertex  and , respectively. In this paper, we study the Hyper-Zagreb index and give upper and lower bounds for .