Revisiting the Diminished Sombor Index

The diminished Sombor index of a graph G with edge set E(G) is defined as DSO(G) = X uv∈E(G) √d2u + d2v du + dv , where du denotes the degree of vertex u. In this paper, we revisit and refine some of the results reported in the recent paper [MATCH Commun. Math. Comput. Chem. 95 (2026) 141–162]. One of the obtained refined results guarantees that DSO(G) decreases when ∗Corresponding author. This work is licensed under a Creative Commons “Attribution 4.0 International” license. 612 any of the edges of G is removed. Also, one of the new results gives the graphs minimizing DSO over the class Gm,n of all connected graphs of order n and size m for 3n ≥ 2m ≥ 2(n + 2). The paper is concluded with an open problem concerning the graphs minimizing DSO over Gm,n for m ≥ max {n + 3, ⌈3n/2⌉}.