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arXiv · 2104.14742

On the vertex-degree based invariants of digraphs

Hanyuan Deng·Jiaxiang Yang·Zikai Tang·Jing Yang·Meiling You
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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.

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Hanyuan Deng, Jiaxiang Yang, Zikai Tang, Jing Yang, Meiling You. 2021-04-30. On the vertex-degree based invariants of digraphs. https://arxiv.org/abs/2104.14742

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