The k-Sombor Index of Trees

For a positive real number k, the k-Sombor index of a graph G, introduced by R ' eti et al, is defined as [GRAPHICS] where d(u) denotes the degree of the vertex u in G. By the definition, SO2(G) is exactly the Sombor index of G, while SO1(G) is the first Zagreb index of G. In this paper, for k >= 1 we present the extremal values of the k-Sombor index of trees with some given parameters, such as matching number, the number of pendant vertices, diameter. This generalizes the relevant results on Sombor index due to Chen, Li and Wang ((2002). Extremal values on the Sombor index of trees, MATCH Communications in Mathematical and in Computer Chemistry, 87, 23-49). Handling SOk(G) appears to be different for k < 1 in contrast to the case when k >= 1. To demonstrate this, we also characterize the extremal trees with respect to the SO1/2 with matching number, the number of pendant vertices and diameter. In addition, three relevant conjectures are proposed.