On the Signless Laplacian Spectral Radius of Digraphs

Abstract

Let $G=(V,E)$ be a digraph with $n$ vertices and $m$ arcs without loops and multiarcs, $V=\{v_1,v_2,\dots ,v_n\}$. Denote the outdegree and average $2$-outdegree of the vertex $v_i$ by $d_i^{+}$ and $m_i^{+}$, respectively. Let $A(G)$ be the adjacency matrix and $D(G)=\text{diag}(d_1^{+},d_2^{+},\dots ,d_n^{+})$ be the diagonal matrix with outdegrees of the vertices of the digraph $G$. Then we call $Q(G)=D(G)+A(G)$ the signless Laplacian matrix of $G$. In this paper, we obtain some upper and lower bounds for the spectral radius of $Q(G)$, which is called the signless Laplacian spectral radius of $G$. We also show that some bounds involving outdegrees and the average $2$-outdegrees of the vertices of $G$ can be obtained from our bounds.