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 C (+)(n) -free if it contains no C (+)(l) ` as a subdigraph for any 2 <= l <= 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 C (+)(n) -free unicyclic digraphs. Moreover, we obtain upper bounds of S-k(G) of C (+)(n) -free digraphs which have vertex-disjoint directed cycles.