Resolving Open Problems on the Euler Sombor Index

Recently, the Euler Sombor index (EUS) was introduced as a novel degree-based topological index. For a graph G, the Euler Sombor index is defined as [Formula Presented] where d_i and d_j denote the degrees of the vertices v_i and v_j, respectively. Very recently, Khanra and Das [Euler Sombor index of trees, unicyclic and chemical graphs, MATCH Commun. Math. Comput. Chem. 94 (2025) 525–548] proposed several open problems concerning the Euler Sombor index. This paper completely resolves two of the most challenging problems posed therein. First, we determine the minimum value of the EUS index among all unicyclic graphs of a fixed order and prescribed girth, and we characterize the extremal graphs that attain this minimum. Building on this result, we further establish the minimum EUS index within the broader class of connected graphs of the same order and girth, and identify the corresponding extremal structures. In addition, we classify all connected graphs that attain the maximum Euler Sombor index (EUS) when both the order and the number of leaves are fixed.