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, \(V = \{v_1, v_2, \ldots, v_n\}\). 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 \(D(G) = \text{diag}(d^+_1, d^+_2, \ldots, d^+_n)\) 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.