Trees with Maximum Vertex-Degree-Based Topological Indices

Let G be a graph with vertex set V (G) = {v(1), v(2),...,v(n)} and edge set E(G), and d(v(i)) be the degree of the vertex v(i). The definition of a vertex-degree-based topological index of G is as follows T-f = T-f(G) = Sigma(vivj is an element of E(G)) f(d(v(i)), d(v(j))), where f(x, y) > 0 is a symmetric real function with x > 0 and y > 0. In this paper, we find the extremal trees with the maximum vertex-degree-based topological index T-f among all trees of order n when f(x, y) is increasing and concave up in respect to variable x (to variable y too, of course).