**2. Applications of Nanostructure and Topological Indices**

In the past few decades, graph theory was widely applied as a tool to study physical and chemical properties of materials. More and more people are interested in this field and as a result chemical graph theory was introduces, and later various topological indices were studied and defined. Moreover, as a combination of chemistry, mathematics and nano science, *nanotechnology* was also studied by means of chemical graph theory. Among these, quantitative structure–activity relationship (QSAR) and quantitative structure-property relationship (QSPR) are analyzed to predict the properties of nanostructure and biological activities. To study QSAR and QSPR, hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index and Zagreb polynomials are applied to predict the bioactivity of nanostructures [12–15].

The Zagreb index is defined to be a topological descriptor which is related to substantial synthetic qualities of the atoms [16]. The particle bond network hyper Zagreb index gives a decent connection to the security of direct dendrimers and also the stretched pharmacies and for processing the strain vitality of cyclo alkanes [17–21]. To relate with some physico-concoction properties, multiple Zagreb index bears much preferred prescient control over the prescient energy of the dendrimers [22–24]. The first and second Zagreb indices were revealed to be used to research the *π*-electron energy of various microscopic particles [25–27].
