On the Vertex-Degree Based Invariants of Digraphs

Let D = (V,A) be a digraph without isolated vertices. A vertex-degree based invariant I(D) related to a real function φ of D is defined as I(D) = 1 2 ∑ uv∈A φ(d + u , d − v ), where du (respectively, du ) denotes the out-degree (respectively, in-degree) of a vertex u. In this paper, we give the extremal values and extremal digraphs of I(D) over all digraphs with n non-isolated vertices. By applying the obtained results, we determine the extremal values of some well-known vertexdegree based topological indices of digraphs, such as the Randić index, the Zagreb indices, the sum-connectivity index, the geometric-arithmetic index, the atom-bond connectivity index and the harmonic index, and characterize the corresponding extremal digraphs.