On the spectral radius and energy of signless Laplacian matrix of digraphs. Issue 3 (March 2022) - Ganie, Hilal A.; Shang, Yilun
Abstract: Let D be a digraph of order n and with a arcs. The signless Laplacian matrix Q ( D ) of D is defined as Q ( D ) = D e g ( D ) + A ( D ), where A ( D ) is the adjacency matrix and D e g ( D ) is the diagonal matrix of vertex out-degrees of D . Among the eigenvalues of Q ( D ) 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. Abstract : Digraphs; Strongly connected digraphs; Signless Laplacian spectral radius; Generalized adjacency spectral radius; Energy