On the extremal Sombor index of trees with a given diameter

Abstract

Based on elementary geometry, Gutman proposed a novel graph invariants called the Sombor index SO(G), which is defined as SO(G)=∑uv∈E(G)dG2(u)+dG2(v), where dG(u) and dG(v) denote the degree of u and v in G, respectively. It has been proved that the Sombor index could predict some physicochemical properties. In this paper, we characterize the extremal graphs with respect to the Sombor index among all the n-order trees with a given diameter. Firstly, we order the trees with respect to the Sombor index among the n-vertex trees with diameter 3. Then, we determine the largest and the second largest Sombor indices of n-vertex trees with a given diameter d≥4 and characterize the corresponding trees. Moreover, for n−d=3, we characterize the extremal n-order trees which reach from the third to the fourth (resp. the sixth, the seventh) largest Sombor indices with d=4 (resp. d=5,d≥6). For n−d≥4, we characterize the extremal n-order trees which reach from the third to the fifth (resp. the eighth, the ninth) largest Sombor indices with d=4 (resp. d=5,d≥6). As consequences, the top four n-order trees with respect to the Sombor index are characterized.