Proof of an open problem on the maximization of the Euler–Sombor index in chemical unicyclic graphs

A topological index is a numerical property of a molecular graph that reflects its structural features. The geometric interpretation and the capacity of topological indices to distinguish between molecular structures have made them an important focus of current research. In this line, numerous degree-based indices have been introduced in recent years. Among these, the Euler–Sombor index, derived from Euler’s approximation formula for the perimeter of an ellipse, has attracted particular attention. For a graph Γ, the Euler–Sombor index (abbreviated as EU–index) is defined as: EU(Γ)=∑ν_iν_j∈E(Γ)d_i²+d_j²+d_id_j, where E(Γ) denotes the edge set and d_i is the degree of a vertex ν_i in Γ. Quite recently, Khanra and Das (2025) posed a problem on characterizing chemical unicyclic graphs with respect to the EU–index in terms of graph order, addressing both the maximizing and minimizing cases. This problem was subsequently discussed by Das et al. (in press), where the minimizing case was completely resolved, while the maximizing case remained open. In this paper, we present a complete characterization of the maximizing problem and identify the corresponding extremal graphs.