On Multiplicative Geometric Arithmetic Index of Some Chemical Structures

Abstract

Graph theoretical study of chemical structures is widely used in physico-chemical characterization of molecules. Several numerical quantities known as topological indices that are associated with the graph structures of molecules have been used as a tool for the characterization of chemical structures. Mathematically, a topological index is a function from any structural property of the graph to a positive real number. With many new developments in chemistry, topological indices are widely used in Quantitative Structure Property Relationship (QSPR) studies. For a molecular graph $G(V,E)$ let $d(u)$ and $d(v)$ denote the degree of vertices $u,v\in V(G) $ respectively and $uv\in E(G)$. The Geometric Arithmetic index is a degree based topological index defined as, $GA_{1}(G)=\sum\limits_{uv\in E(G)}\frac{\sqrt{d(u)d(v)}}{\frac{d(u)+d(v)}{2}}$. Its multiplicative version, called the Multiplicative Geometric Arithmetic index was introduced recently which is defined as, $GAII(G)=\prod\limits_{uv\in E(G)}\frac{\sqrt{d(u)d(v)}}{\frac{d(u)+d(v)}{2}}$. In this paper, we determine the exact value of \textit{GAII} for some common polycyclic organic structures, namely graphene, triangular benzenoids, benzenoid series, benzenoid systems and phenylenes. Further we obtain a comparison between $GA_{1}$ and $GAII$ indices.