On the spectral radius and energy of signless Laplacian matrix of digraphs

Abstract

Let D be a digraph of order n and with a arcs. The signless Laplacian matrix $Q\left(D\right)$ of D is defined as $Q\left(D\right)=Deg\left(D\right)+A\left(D\right)$ , where $A\left(D\right)$ is the adjacency matrix and $Deg\left(D\right)$ is the diagonal matrix of vertex out-degrees of D. Among the eigenvalues of $Q\left(D\right)$ the eigenvalue with largest modulus is the signless Laplacian spectral radius or the Q-spectral radius of D. The main contribution of this paper is a series of new lower bounds for the Q-spectral radius in terms of the number of vertices n, the number of arcs, the vertex out-degrees, the number of closed walks of length 2 of the digraph D. We characterize the extremal digraphs attaining these bounds. Further, as applications we obtain some bounds for the signless Laplacian energy of a digraph D and characterize the extremal digraphs for these bounds.

Keywords: Digraphs; Energy; Generalized adjacency spectral radius; Signless Laplacian spectral radius; Strongly connected digraphs.