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Let $G=(V(G) ,E(G))$ be a digraph without loops and multiarcs, where$V(G)=\{v_1,v_2,\ldots,v_n\}$ and $E(G)$ are the vertex set and the arc set of$G$, respectively. Let $d_i^{+}$ be the outdegree of the vertex $v_i$. Let$A(G)$ be the adjacency matrix of $G$ and$D(G)=\textrm{diag}(d_1^{+},d_2^{+},\ldots,d_n^{+})$ be the diagonal matrixwith outdegrees of the vertices of $G$. Then we call $Q(G)=D(G)+A(G)$ thesignless Laplacian matrix of $G$. The spectral radius of $Q(G)$ is called thesignless 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.

Some upper bounds for the signless Laplacian spectral radius of digraphs