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, …,vn} and E(G) are the‎ ‎vertex set and the arc set of G‎, ‎respectively‎. ‎Let d+i be the‎ ‎outdegree of the vertex vi‎. ‎Let A(G) be the adjacency matrix of‎ ‎G and D(G)=diag(d+1,d+2,…,d+n) be the‎ ‎diagonal matrix with outdegrees of the vertices of G‎. ‎Then we call‎ ‎Q(G)=D(G)+A(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.