RETRACTED ARTICLE: Lower bound for the Sombor index of trees with a given total domination number

Recently, finding extremal structures of graphs on Sombor index has received a lot of attention. The Sombor (SO) index of a graph G is defined by the sum of weights \(\sqrt{deg_{G}(u)^{2}+deg_{G}(v)^{2}}\) $\sqrt{de{g}_G{(u)}^2+de{g}_G{(v)}^2}$ over all edges uv of G, where \(deg_{G}(u)\) $de{g}_G(u)$ stands for the degree of vertex u in G. In this article, we obtain a lower bound on Sombor index of trees with a given order and total domination number, and characterize the trees achieving the bound.