Vertex-degree-based topological indices of oriented trees

Let D be a digraph with arc set A(D). A vertex-degree-based topological index phi is defined in D as phi (D) = 1/2 sigma(uv is an element of A(D)) phi(+)(du) ,dv(-), where d(u)(+) is the outdegree of vertex u , d(v)(-) is the indegree of vertex v, and phi(x,y) is a (symmetric) function. We study in this paper the extremal value problem of a VDB topological index phi over the set of orientations of a tree T . We show that one extreme value is attained in sink-source orientations, and when the tree has no adjacent branching vertices, the other extremal value occurs in balanced orientations. In the case the tree has adjacent branching vertices, considering the double-star tree, we show that a VDB topological index phi may not be invariant over the set of balanced orientations, and the extremal value can occur in non-balanced orientations. (c) 2022 Elsevier Inc. All rights reserved.