Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological index. The exponential atom-bond connectivity index is defined as follows: eABC=eABC(Υ)=∑vivj∈E(Υ)edi+dj−2didj, where di is the degree of the vertex vi in Υ. In this paper, we prove that the double star DSn−3,1 is the second maximal graph with respect to the eABC index of trees of order n. We give an upper bound on eABC of unicyclic graphs of order n and characterize the maximal graphs. The graph K1∨(P3∪(n−4)K1) gives the maximal graph with respect to the eABC index of bicyclic graphs of order n. We present several relations between eABC(Υ) and ABC(Υ) of graph Υ. Finally, we provide a conclusion summarizing our findings and discuss potential directions for future research.