The Path and the Star as Extremal Values of Vertex-Degree-Based Topological Indices Among Trees

We denote by T-n the set of trees with n vertices, P-n is an element of T-n is the path tree and S-n is an element of T-n is the star tree. Let phi be a vertex-degree-based topological index defined over T-n. For any tree T is an element of T-n, we find an expression of phi (T) in terms of phi (P-n) and a function f(phi) associated to phi, in such a way that the derivatives of f(phi) over a compact set gives information on when the path P-n is an extremal value of phi over T-n, for n >= 3. Similarly, we present results which give information on when the star S-n is an extremal value of phi over T-n, for n >= 3. As an application, we determine extremal trees for exponential vertex-degree-based topological indices.