Abstract
Let \(G\) be a graph with vertex set \(V(G)=\{v_1, v_2, \dots, v_n\}\) 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 \[ \mathcal{T}_f=\mathcal{T}_f(G)=\sum_{v_iv_j\in
E(G)}f(d(v_i),d(v_j)), \] 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 \(\mathcal{T}_f\) 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).