General \((\alpha,2)\) Path Sum Connectivity Index of Nanostructures

The general path sum connectivity index of a molecular graph, denoted as \(^{t}\chi_{\alpha}(G)\), is defined for a graph \(G\), where \(\alpha\) is a positive real number and \(t\) is a positive integer. This index is expressed as:
\[
^{t}\chi_{\alpha}(G) = \sum_{p^{t} = v_{j_{1}}v_{j_{2}}\dots v_{j_{t+1}} \subseteq G} \left[ d_{G}(v_{j_{1}}) + d_{G}(v_{j_{2}}) + \dots + d_{G}(v_{j_{t+1}}) \right]^{\alpha},
\]
where \(p^t\) represents a path of length \(t\) within the graph. In this work, we compute the general path sum connectivity index for various nanostructures, including phenylene, naphthalene, anthracene, and tetracene nanotubes. This index is particularly useful in investigating the physico-chemical properties of chemical compounds and plays a crucial role in the analysis of three-dimensional quantitative structure-activity relationships (3D-QSAR) and molecular chirality.