Vertex-degree function index on oriented graphs

Abstract

Let D be a digraph with vertex set V(D) and arc set A(D). For a real function f defined on nonnegative real numbers, the vertex-degree function index $H_{f}(D)$ of the digraph D is defined as

$$\begin{aligned} H_{f}(D)=\frac{1}{2}\sum _{u\in V(D)}\left[ f(d_{u}^{+}) +f(d_{u}^{-}) \right] , \end{aligned}$$

where $d_u^+$ and $d_u^-$ denote the outdegree and the indegree of u, respectively. In this paper we find the extremal values of $H_{f}$ among orientations of a given graph G, when f is a convex (or concave) real function on $\left[ 0,+\infty \right)$.