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