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

초록

The general Sombor (SO alpha) index of a graph G is defined as the sum of weights (d(x)(2)(G)+d(y)(2)(G))(alpha) over all edges xy of G, where alpha not equal 0 is a real number and d(x)(G) denotes the degree of a vertex x in G. In this paper, we focus on two specific classes of trees: T-n,T-b, the set of all n-vertex trees with b branching vertices, and T-n,T-Delta, the set of all n-vertex trees with prescribed maximum degree Delta . Thus the purpose of this paper is twofold concerning the SO alpha index: (i) to characterize the minimal trees in T-n,T-b when alpha>0, and (ii) to characterize the maximal trees in T-n,T-Delta when 0<alpha<1. The results of (i) hold true even when the class T-n,T-b 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 SO alpha (0<alpha<1) index and, moreover, establish analogous results for the well-known general sum-connectivity index, thereby addressing the corresponding unresolved cases for both indices.

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