On the sum of the k largest absolute values of Laplacian eigenvalues of digraphs

Let $L(G)$ be the Laplacian matrix of a digraph $G$ and $S_k(G)$ sum $k$ largest absolute values eigenvalues $G$. $C_n^+$ with $n+1$ vertices obtained from directed cycle $C_n$ by attaching pendant arc whose tail is on $C_n$. A $\mathbb{C}_n^+$-free if it contains no $C_{\ell}^+$ as subdigraph for any $2\leq \ell \leq n-1$. In this paper, we present lower bounds $S_n(G)$ digraphs order $n$. We provide exact cycles unicyclic digraphs. Moreover, obtain upper which have vertex-disjoint cycles.