The diminished Sombor index is a degree-based topological index that normalizes the Sombor contribution of each edge (defined as the Euclidean norm of the endpoint degrees) by the sum of those degrees, thereby making the index independent of graph size and ensuring a more balanced reflection of the relative degree contributions of adjacent vertices. In this paper, we investigate the extremal behavior of the diminished Sombor index over the class of connected bipartite graphs with fixed order and diameter. We establish a sharp upper bound for this index within the family of all bipartite graphs on a given number of vertices and with a prescribed diameter, and we completely characterize the extremal graphs attaining this bound. Furthermore, we prove that the maximum diminished Sombor index strictly decreases as the diameter increases. As a consequence, we determine the connected bipartite graphs of fixed order that achieve the three largest values of the diminished Sombor index.
Wazzan et al. (Thu,) studied this question.