Some upper bounds for the signless Laplacian spectral radius of digraphs

Abstract

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 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.