On the exponential augmented Zagreb index of graphs

A topological index is a numerical descriptor in mathematical chemistry and graph theory that quantifies a molecule’s structural properties without considering its three-dimensional arrangement. A crucial factor to consider when exploring topological indices is their capacity to distinguish between different structures. In light of this, the exponential vertex-degree-based topological index is put forward in the literature. The present work focuses on investigating the mathematical properties and application potential of the exponential augmented Zagreb index (EAZ). The EAZ index for a graph \(\Upsilon \) $\mathrm{{\rm Y}}$ is defined as \(\begin{aligned} EAZ(\Upsilon )=\sum \limits _{v_iv_j \in E(\Upsilon )}\,e^{\displaystyle {\left( \frac{{d_i\,d_j}}{d_i+d_j-2}\right) ^{3}}}, \end{aligned}\) $$\begin{array}{c}\hfill EAZ(\mathrm{{\rm Y}})=\sum _{v_i{v}_j\in E(\mathrm{{\rm Y}})}{e}^{{\displaystyle {\left(\frac{d_i{d}_j}{d_i+d_j-2}\right)}^3}},\end{array}$$ where \(d_i\) $d_i$ represents the degree of a vertex \(v_i\) $v_i$ . Crucial upper and lower bounds of EAZ for numerous classes of graphs like bipartite, unicyclic, bicyclic, chemical graph, and general graphs are derived. The bounds are computed in terms of different graph parameters including graph order, size, maximum degree, number of pendant vertices and independence number. The extremal graphs for which the bounds appear are also characterized. Moreover, the EAZ index is found to correlate well with some physico-chemical properties of octanes.