Topological indices are numerical descriptors derived from the structure of a molecular graph and provide valuable information about the chemical, physical, and biological properties of molecules. The diminished Sombor index (DSO) of a graph G is defined as \( DSO(G)=\sum \limits _{v_i v_j \in E(G)} \frac{\sqrt{d\left( v_i\right) ^2+d\left( v_j\right) ^2}}{d\left( v_i\right) +d\left( v_j\right) }, \) where \(d(v_i)\) denotes the degree of the vertex \(v_i\) and E(G) is the edge set of G. The characterization of extremal chemical trees with respect to topological indices is a well-established problem in chemical graph theory. Cruz et al. [1] provided a set of sufficient conditions for such characterizations. While most degree-based indices satisfy these conditions, the DSO index does not, making the problem of identifying extremal chemical trees particularly challenging. In this work, we completely resolve this problem and further investigate the applicability of DSO in structure–property modeling using chemical tree datasets. The results demonstrate that DSO exhibits strong predictive performance for enthalpy of vaporization, outperforming several well-known degree-based indices in both direct evaluation and 5-fold cross-validation analyses. These findings strengthen the claims made by Movahedi et al. [2].
