Sharp bounds for the signless Laplacian spectral radius of digraphs

Let G=(V,E) be a digraph with n vertices and m arcs without loops and multiarcs, and vertex set V={^v1,^v2,...,^vn}. Denote the outdegree and average 2-outdegree of the vertex ^vi by di+ and mi+, respectively. Let A(G) be the adjacency matrix and D(G)=diagd1+,d2+,... ,dn+ be the diagonal matrix with outdegree of the vertices of the digraph G. Then we call Q(G)=D(G)+A(G) the signless Laplacian matrix of G. Let q(G) denote the signless Laplacian spectral radius of the digraph G. In this paper, we present several improved bounds in terms of outdegree and average 2-outdegree for the signless Laplacian spectral radius of digraphs. Then we give an example to compare the bounds.