On the vertex-degree based invariants of digraphs

Abstract

Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $\phi$ of $D$ is defined as a summation over all arcs, $I(D)=\frac{1}{2}{\sum }_{uv\in A}\phi (d_u^{+},d_v^{-})$, where $d_u^{+}$ (resp. $d_u^{-}$) denotes the out-degree (resp. 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. Applying these results, we obtain the extremal values of some vertex-degree based topological indices of digraphs, such as the Randić index, the Zagreb index, the sum-connectivity index, the $GA$ index, the $ABC$ index and the harmonic index, and the corresponding extremal digraphs.