Sharp Upper and Lower Bounds of VDB Topological Indices of Digraphs

A vertex-degree-based (VDB, for short) topological index f induced by the numbers {f(ij)} was recently defined for a digraph D, as phi D=1/2 n-ary sumation(uv)f(du+dv-), where d(u)(+) denotes the out-degree of the vertex u, d(v)(-) denotes the in-degree of the vertex v, and the sum runs over the set of arcs uv of D. This definition generalizes the concept of a VDB topological index of a graph. In a general setting, we find sharp lower and upper bounds of a symmetric VDB topological index over D-n, the set of all digraphs with n non-isolated vertices. Applications to well-known topological indices are deduced. We also determine extremal values of symmetric VDB topological indices over OTn and OG, the set of oriented trees with n vertices, and the set of all orientations of a fixed graph G, respectively.