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) =ₔₕ ₄ (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.