Sharp bounds for the signless Laplacian spectral radius of digraphs

Let G=(V,E)G=(V,E) be a digraph with n vertices and m   arcs without loops and multiarcs, and vertex set V={v1,v2,…,vn}V={v1,v2,…,vn}. Denote the outdegree and average 2-outdegree of the vertex vivi by di+ and mi+, respectively. Let A(G)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)Q(G)=D(G)+A(G) the signless Laplacian matrix of G  . Let q(G)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.