Vertex-degree-based topological indices of oriented trees

Abstract

Let D be a digraph with arc set A(D). A vertex-degree-based topological index φ is defined in D asφ(D)=12∑uv∈A(D)φdu+,dv−,where du+ is the outdegree of vertex u, dv− is the indegree of vertex v, and φx,y is a (symmetric) function. We study in this paper the extremal value problem of a VDB topological index φ 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 φ may not be invariant over the set of balanced orientations, and the extremal value can occur in non-balanced orientations.