General Sombor index: a study of branching in trees and solution for maximal trees with prescribed maximum degree

The general Sombor () index of a graph G is defined as the sum of weights over all edges xy of G, where is a real number and denotes the degree of a vertex x in G. In this paper, we focus on two specific classes of trees:, the set of all n-vertex trees with b branching vertices, and, the set of all n-vertex trees with prescribed maximum degree. Thus the purpose of this paper is twofold concerning the index: (i) to characterize the minimal trees in when, and (ii) to characterize the maximal trees in when. The results of (i) hold true even when the class is confined to the class of chemical trees and also recover previously known results for the Sombor index. The findings in (ii) resolve a previously posed problem for the index and, moreover, establish analogous results for the well-known general sum-connectivity index, thereby addressing the corresponding unresolved cases for both indices.