AFFECTION OF MOORE–PENROSE GENERALIZED INVERSE ON MATRICES OF CUBIC COMPLETE GRAPH AND NON-EMPTY REGULAR (COMPLETE) GRAPH

Abstract

Background: The Moore-Penrose generalized inverse of a matrix was introduced by E. H. Moore (Adi Ben-Israel, 2003) as the "general reciprocal" of a matrix, and by Penrose (Ali Aziz Ali, 1983) as the "generalized inverse" of a matrix. This generalized inverse exists and is unique for every matrix (not necessarily square) with complex elements. The definitions employed by Moore - Penrose, although not identical, have been shown to be equivalent by Rado.                                

Methodology and aims: The aim of this paper is to discuss some new properties of a non- empty regular graph, and non- empty cubic complete regular graph and some properties of matrices of an incidence matrix of . Also, we use the affection of the Moore–Penrose Generalized Inverse (MPGI) of matrix  (Adi Ben-Israel, 2003, B. Noble, 1976, S. L. Campbell, C. D. Meyer, 1979, Li Weiguo, et al., 2013), we get to    , and we introduced some new theorems and results.

Results and Discussion: Let  be a non–empty cubic regular graph, then . An element  of the matrix  is the product of the two rows  of the matrix . That is  in the matrix  is  in the graph, that is, . Additionally, If the number of vertices are even, then the resulting the graph are discrete (disjoint) graph such that number of the Composites equals to  , where .

Conclusion: Our hope that this paper will motivate further study of affection MPGI of matrices of regular graphs when . Also on complete graphs when , and on all graphs in general.