Open problems on the exponential vertex-degree-based topological indices of graphs

Several topological indices (second Zagreb index, augmented Zagreb index, atom-bond connectivity index etc.) are possibly the graph-based molecular structure descriptors most widely applied in chemistry and pharmacology. One important aspect in the study of topological indices is their discrimination ability. In view of this, the exponential vertex-degree-based topological index was introduced in the literature. Cruz et al. (2020) mentioned some open problems on the exponential vertex-degree-based topological indices of trees. In this paper, we solve two open problems on the exponential second Zagreb index and the exponential augmented Zagreb index of trees. Moreover, we present some bounds on the exponential atom-bond connectivity index of graphs and characterize the extremal graphs.