# Sufficient conditions on the zeroth-order general Randic index for maximally edge-connected digraphs

Document Type : Original paper

Author

RWTH Aachen University

Abstract

Let $D$ be a finite and simple digraph with vertex set $V(D)$. For a vertex $v\in V(D)$, the degree of $v$, denoted by $d(v)$, is defined as the minimum value of its out-degree $d^+(v)$ and its in-degree $d^-(v)$. Now let $D$ be a digraph with minimum degree $\delta\ge 1$ and edge-connectivity $\lambda$. If $\alpha$ is real number, then, analogously to graphs, we define the zeroth-order general Randi'{c} index by $\sum_{x\in V(D)}(d(x))^{\alpha}$. A digraph is maximally edge-connected if $\lambda=\delta$. In this paper, we present sufficient conditions for digraphs to be maximally edge-connected in terms of the zeroth-order general Randi'{c} index, the order and the minimum degree when $\alpha <0$, $0<\alpha <1$ or $1<\alpha\le 2$. Using the associated digraph of a graph, we show that our results include some corresponding known results on graphs.

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