The sharp lower bound of tricyclic graphs with respect to the ISI index: applications in octane isomers and benzenoid hydrocarbons

The inverse sum indeg index of a given graph G is symbolized with ISI and defined by sum of the weights \(\frac{d_u d_v}{d_u + d_v}\) entire links \(uv\in G\). We denote \(d_u\) (resp. \(d_v\)) the degree of a vertex u (resp. v) of G. In this paper, we obtained the sharp lower bound of tricyclic graphs with respect to the ISI index of order \(n\geqslant 6\) with size \(n + 2\). By using atoms as vertices and chemical bonds as edges, graph theory permits the representation of molecular structures as mathematical entities called graphs. Based on the above concept, we formulated the ISI index of octane isomers (OI) and benzenoid hydrocarbons (BH) and compared the values of ISI index with various degree-based TI’s via their correlations and chemical properties. The structural analysis of octane isomers is an important application of this research, as the ISI index delivers insights into stability designs across various isomeric forms.