A normal quotient analysis for some families of oriented four-valent graphs

Topological indices are numerical descriptors that describe the chemical structures of chemical compounds using their molecular graphs. Recent advancements in topological indices have seen the emergence of neighborhood indices and entire topological indices, offering distinct perspectives on molecular structure. Neighborhood indices emphasize local atomic environments, while entire indices provide a comprehensive view by considering interactions between atoms, bonds, and their combinations. To achieve a more balanced and informative representation, we introduce 'entire neighborhood indices'. By integrating the localized focus of neighborhood indices within the framework of entire indices, these new descriptors offer a more complete picture of molecular structure and are expected to significantly enhance the accuracy of predictions for various molecular properties. In this paper, we introduce a new version of Zagreb topological indices named first, second, and modified entire neighborhood topological indices; denoted by \(NM_{1}^{\varepsilon}\), \(NM_{2}^{\varepsilon}\), and \(MNM_{1}^{\varepsilon}\), respectively. The structure-property regression analysis is used to investigate and compute the chemical significant of these newly introduced indices for the prediction of the physico-chemical properties of octane isomers and benzenoid hydrocarbons benchmark datasets. We analays and calculate the specific formulae of the entire neighborhood indices for several important graph families such as path, regular, cycle, complete, bipartite, book, gear and helm graph. Furthermore, we determine the exact value of these new indices for some types of bridge graphs and Sierpinski graphs.

Let R be a prime ring with center Z ( R ). A map G : R → R is called a multiplicative (generalized) ( α , β )-derivation if G ( xy )= G ( x ) α ( y )+ β ( x ) g ( y ) is fulfilled for all x ; y ∈ R , where g : R → R is any map (not necessarily derivation) and α ; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) ( α , β )-derivations associated with the mappings g and h , respectively, on R and α , β are automorphisms of R . The main objective of the present paper is to investigate the following algebraic identities: ( i ) G ( xy ) + α ( xy ) = 0, ( ii ) G ( xy ) + α ( yx ) = 0, ( iii ) G ( xy ) + G ( x ) G ( y ) = 0, ( iv ) G ( xy ) = α ( y ) ○ H ( x ) and ( v ) G ( xy ) = [ α ( y ), H ( x )] for all x , y in an appropriate subset of R .

Let R be a ring with center Z and alpha, beta and d mappings of R. A mapping F of R is called a centrally-extended multiplicative (generalized)-(alpha, beta)-derivation associated with d if F(xy) - F(x)alpha(y) - beta(x)d(y) is an element of Z for all x, y is an element of R. The objective of the present paper is to study the following conditions: (i) F(xy) +/- beta(x)G(y) is an element of Z, (ii) F(xy) +/- g(x)alpha(y) is an element of Z and (iii) F(xy) +/- g(y)alpha(x) is an element of Z for all x, y in some appropriate subsets of R, where G is a multiplicative (generalized)-(alpha, beta)-derivation of R associated with the map g on R.