Extremal Laplacian energy of directed trees, unicyclic digraphs and bicyclic digraphs

Let A(G) and D⁺(G) be the adjacency matrix of a digraph G with n vertices and the diagonal matrix of vertex outdegrees of G, respectively. Then the Laplacian matrix of the digraph G is L(G)=D⁺(G)−A(G). The Laplacian energy of a digraph G is defined as LE(G)=∑_i=1 ⁿλ_i ² by using second spectral moment, where λ₁,λ₂,…,λ_n are all the eigenvalues of L(G) of G. In this paper, by using arc shifting operation and out-star shifting operation, we determine the directed trees, unicyclic digraphs and bicyclic digraphs which attain maximal and minimal Laplacian energy among all digraphs with n vertices, respectively.