On the Diminished Sombor Index of Fixed-Order Molecular Graphs With Cyclomatic Number at Least 3

For a graph G with edge set E, let $d(u)$ denote the degree of a vertex u in G. The diminished Sombor (DSO) index of G is defined as DSO(G)=∑uv∈ E√(d(u))²+(d(v))²(d(u)+d(v))⁻¹. The cyclomatic number of a graph is the smallest number of edges whose removal makes the graph acyclic. A connected graph of maximum degree at most $4$ is known as a molecular graph. The primary motivation of the present study comes from a conjecture concerning the minimum DSO index of fixed-order connected graphs with cyclomatic number $3$, posed in the recent paper [F. Movahedi, I. Gutman, I. Redžepović, B. Furtula, Diminished Sombor index, MATCH Commun. Comput. Chem. 95 (2026) 141--162]. The present paper gives all graphs minimizing the DSO index among all molecular graphs of order n with cyclomatic number , provided that n≥ 2(-1)≥4.