Some upper bounds for the signless laplacian spectral radius of digraphs

Let G = (V(G), E(G) be a digraph without loops and multiarcs, where V(G) = (v₁, v₂,..., v_n) and E(G) are the vertex set and the arc set of G, respectively. Let be the outdegree of the vertex v_i. Let A(G) be the adjacency matrix of G and D(G) = diag be the diagonal matrix with outdegrees of the vertices of G. Then we call Q(G) = D(G) +-4(G) the signless Laplacian matrix of G. The spectral radius of Q(G) is called the signless Laplacian spectral radius of G. denoted by q(G). In this paper, some upper bounds for q(G) are obtained. Furthermore, some upper bounds on q(G) involving outdegrees and the average 2-outdegrees of the vertices of G are also derived.