Degree-Ratio Sombor Index
The degree-ratio Sombor (DRSO) index is a recently introduced variant of the extensively studied Sombor index. In this paper, we establish an upper bound on the DRSO index in terms of the size, minimum degree, and maximum degree of a graph, which provides the unique graph maximizing the DRSO index among all fixed-order trees. Also, we give an upper bound for the DRSO index in terms of the order, size, and maximum degree of a graph, from which it follows that the star graph uniquely maximizes the DRSO index over the class of all fixed-order connected graphs. In addition, we rectify two results from [MATCH Commun. Math. Comput. Chem. 97 (2027) 135–164], one related to a lower bound on the DRSO index and the other concerning extremal values of the DRSO index among all fixed-order connected graphs. We characterize the graphs that maximize the DRSO index among all fixed-order (i) unicyclic graphs, (ii) trees with a prescribed number of pendent vertices, and (iii) trees with a fixed matching number. Furthermore, we determine the structure of graphs that minimize the DRSO index among all fixed-order k-cyclic graphs, for k ≥ 1 and sufficiently large values of the order.