Modified Sombor index of trees

Abstract

The modified Sombor index of a graph G, denoted by ᵐSO(G), is defined as the sum of weights 1/√d2G(u)+d2G(v) of all edges uv of E(G), where dG(u) denotes the degree of a vertex u in G. In this paper we show that for any tree T of order n with maximum degree ∆, ᵐSO(T) ≤ ∆/√∆2 + 4 + (n − 2∆ − 1)/√8 + ∆/√5, when ∆ ≤ n−1/2 and ᵐSO(T) ≤ (2∆ + 1 − n)/√∆2 + 1 + (n − ∆ − 1)/√∆2 + 4 + (n − ∆ − 1)/√5, when ∆ > n−1/2. Also we determine the extremal trees achieve these bounds.