On harmonic index of trees with a given total domination number

Harmonic index is one of the variants of the well-known Randi & cacute; index. For a graph G, the harmonic index of G is defined by the sum of weights 2 d(u)+d(v) over all edges uv of G, where d(u) stands for the degree of vertex u in G. In this paper, an upper bound on the harmonic index of trees with a given order and total domination number is determined and the corresponding extremal trees are characterized. Moreover, our conclusions correct the results presented by Hasni et al., Sharp upper bound for harmonic index of trees with given total domination number, Ars Combin. 159 (2024) 179-186.