The Path and the Star as Extremal Values of Vertex–Degree–Based Topological Indices Among Trees (2019)
We denote by Tn the set of trees with n vertices, Pn ∈ Tn is the path tree and Sn ∈ Tn is the star tree. Let ϕ be a vertex-degree-based topological index defined over Tn. For any tree T ∈ Tn, we find an expression of ϕ (T) in terms of ϕ (Pn) and a function fϕ associated to ϕ, in such a way that the derivatives of fϕ over a compact set gives information on when the path Pn is an extremal value of ϕ over Tn, for n ≥ 3. Similarly, we present results which give information on when the star Sn is an extremal value of ϕ over Tn, for n ≥ 3. As an application, we determine extremal trees for exponential vertex-degree-based topological indices.