On EAZ index of unicyclic and bicyclic graphs, general graphs in terms of the number of cut edges

A topological index, derived from the graph representation of a molecule, condenses structural information and is instrumental in predicting chemical and physical properties. An essential consideration in the investigation of topological indices is their ability to differentiate among various structures. This fact leads to the development of exponential degree-based topological indices. The central theme of this work is the exponential augmented Zagreb index (EAZ), which was observed to exhibit strong correlations with numerous properties of octanes. The EAZ index for a graph \(\Gamma \) $\mathrm{\Gamma }$ is defined as \(EAZ(\Gamma )= \sum \nolimits _{v_{i}v_{j} \in E(\Gamma )} F(\delta _{i},\delta _{j})\) $EAZ(\mathrm{\Gamma })={\sum }_{v_i{v}_j\in E(\mathrm{\Gamma })}F({\delta }_i,{\delta }_j)$ , where \(\delta _i\) ${\delta }_i$ represents the degree of a vertex \(v_i\) $v_i$ and \(F(x, y)=e^{\left( \frac{xy}{x+y-2}\right) ^{3}}\) $F(x,y)=e^{{\left(\frac{\mathit{xy}}{x+y-2}\right)}^3}$ . We intend to establish tight bounds of EAZ with determining extremal graphs. Our systematic investigation offers maximal bicyclic graph in terms of graph order p. We also find minimal unicyclic graph when both p and girth are specified. In addition, the minimal graph for EAZ is characterized with respect to number of cut edges and p.