Resolving an Open Problem on the Exponential Arithmetic-Geometric Index of Unicyclic Graphs

Recently the exponential arithmetic-geometric index ($EAG$) was introduced. The exponential arithmetic-geometric index ($EAG$) of a graph $G$ is defined as $$EAG(G)=\sum_{v_iv_j \in E(G)}\,e^\frac{d_i + d_j}{2 \sqrt{d_i d_j}},$$ where $d_i$ represents the degree of the vertex $v_i$ in $G$. The characterization of extreme structures in relation to graph invariants from the class of unicyclic graphs is an important problem in discrete mathematics. Cruz, Rada and Sanchez [Extremal unicyclic graphs with respect to vertex-degree-based topological indices, {\it MATCH Commun. Math. Comput. Chem.\/} {\bf 88} (2022) 481--503] proposed a unified method for finding extremal unicyclic graphs for exponential degree-based graph invariants. However, in the case of $EAG$, this method is insufficient to generate the maximal unicyclic graph. Consequently, the same article presented an open problem for the investigation of the maximal unicyclic graph with respect to this invariant. This article completely characterizes the maximal unicyclic graph in relation to $EAG$.