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)=∑vivj∈E(G)edi+dj2didj, where di represents the degree of the vertex vi 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 et al., 2022 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 for generating 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.