On the spectral radius and energy of signless Laplacian matrix of digraphs.

Let be a digraph of order and with arcs. The signless Laplacian matrix of is defined as , where is the adjacency matrix and is the diagonal matrix of vertex out-degrees of . Among the eigenvalues of the eigenvalue with largest modulus is the signless Laplacian spectral radius or the -spectral radius of . The main contribution of this paper is a series of new lower bounds for the -spectral radius in terms of the number of vertices , the number of arcs, the vertex out-degrees, the number of closed walks of length 2 of the digraph . We characterize the extremal digraphs attaining these bounds. Further, as applications we obtain some bounds for the signless Laplacian energy of a digraph and characterize the extremal digraphs for these bounds.