Open problem on the maximum exponential augmented Zagreb index of unicyclic graphs

Abstract

A topological index is a numerical property of a molecular graph that explains structural features of molecules. The potential of topological indices to discriminate between distinct structures is a significant topic to investigate. In this context, the exponential degree-based indices were put forward in the literature. The present work focuses on the exponential augmented Zagreb index (EAZ), which is defined for a graph G as

$$\begin{aligned} EAZ(G)=\sum \limits _{v_{i}v_{j} \in E(G)}\,e^{\displaystyle {\left( \frac{{d_i\,d_j}}{d_i+d_j-2}\right) ^{3}}}, \end{aligned}$$

where \(d_i\) represents the degree of the vertex \(v_i\)and E(G) denotes the edge set of G. This work characterizes the maximal unicyclic graph for EAZ in terms of graph order, which was posed as an open problem in the recent article Cruz et al. (MATCH Commun Math Comput Chem 88:481-503, 2022).