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

The general Sombor ( $$\mathcal{S}\mathcal{O}_\alpha $$ ) index of a graph G is defined as the sum of weights $$\Big (d^2_x(G) +d^2_y(G)\Big )^\alpha $$ over all edges xy of G, where $$\alpha \ne 0$$ 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: $${{\mathcal {T}}}_{n,b}$$ , the set of all n-vertex trees with b branching vertices, and $${{\mathcal {T}}}_{n,\Delta }$$ , the set of all n-vertex trees with prescribed maximum degree $$\Delta $$ . Thus the purpose of this paper is twofold concerning the $$\mathcal{S}\mathcal{O}_\alpha $$ index: (i) to characterize the minimal trees in $${{\mathcal {T}}}_{n,b}$$ when $$\alpha > 0$$ , and (ii) to characterize the maximal trees in $${{\mathcal {T}}}_{n,\Delta }$$ when $$0