Abstract
Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $\varphi$ of $D$ is defined as a summation over all arcs, $I(D) = \frac{1}{2}\sum_{uv\in A}{\varphi(d_u^+,d_v^-)}$, 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 Randi\'{c} index, the Zagreb index, the sum-connectivity index, the $GA$ index, the $ABC$ index and the harmonic index, and the corresponding extremal digraphs.
Highlights
A digraph D = (V, A) is an ordered pair (V, A) consisting of a non-empty finite set V of vertices and a finite set A of ordered pairs of distinct vertices called arcs
If a ∈ A is an arc from vertex u to vertex v, we indicate this by writing a = uv
Rada [7] extended the concept of vertex-degree based topological indices of graphs to digraphs. They obtained the extremal values of the Randic index of digraphs over Dn, and found the extremal values of the Randic index over the set of all oriented trees with n vertices
Summary
Rada [7] extended the concept of vertex-degree based topological indices of graphs to digraphs They obtained the extremal values of the Randic index of digraphs over Dn, and found the extremal values of the Randic index over the set of all oriented trees with n vertices. The digraph obtained from the star on n vertices by replacing each of its edges with a pair of symmetric arcs satisfies (5). =0 and pij = 0 for all (i, j) ∈ S2, i.e., D is the digraph obtained from Kn by replacing each edge with a pair of symmetric arcs. Theorems 1-3 show that the results on digraphs are different from the results on graphs in [2]
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