Modified Zagreb Connection Indices for Benes Network and Related Classes

Abstract

The study of networks such as Butterfly networks, Benes networks, interconnection networks, David-derived networks through graph theoretical parameters is among the modern trends in the area of graph theory. Among these graph theoretical tools, the topological Indices $\left(\text{TIs}\right)$ have been frequently used as graph invariants. $\text{TIs}$ are also the essential tools for quantitative structure activity relationship (QSAR) as well as quantity structure property relationships (QSPR). TIs depend on different parameters, such as degree and distance of vertices in graphs. The current work is devoted to the derivation of 2-distance based TIs, known as, modified first Zagreb connection index ${\text{ZC}}_1^{\ast }$ and first Zagreb connection index $\left({\text{ZC}}_1\right)$ for $r-$ dimensional Benes network and some classes generated from Benes network. The horizontal cylindrical Benes network $\left(\text{HCB}\left(r\right)\right)$ , vertical cylindrical Benes network $\left(\text{VCB}\left(r\right)\right)$ , and toroidal Benes network $\left(\text{TB}\left(r\right)\right)$ are the three classes generated by identifying the vertices of the first row with the last row, the first column with the last column of the Benes network. The obtained results are also analyzed through graphical tools.