The k-Sombor Index of Trees

Abstract

For a positive real number [Formula: see text], the [Formula: see text]-Sombor index of a graph [Formula: see text], introduced by Réti et al, is defined as [Formula: see text] where [Formula: see text] denotes the degree of the vertex [Formula: see text] in [Formula: see text]. By the definition, [Formula: see text] is exactly the Sombor index of [Formula: see text], while [Formula: see text] is the first Zagreb index of [Formula: see text]. In this paper, for [Formula: see text] we present the extremal values of the [Formula: see text]-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 [Formula: see text] appears to be different for [Formula: see text] in contrast to the case when [Formula: see text]. To demonstrate this, we also characterize the extremal trees with respect to the [Formula: see text] with matching number, the number of pendant vertices and diameter. In addition, three relevant conjectures are proposed.