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

Ganie, Hilal A. and Shang, Yilun (2022) On the spectral radius and energy of signless Laplacian matrix of digraphs. Heliyon, 8 (3). e09186. ISSN 2405-8440

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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)=Deg(D)+A(D), where A(D) is the adjacency matrix and Deg(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.

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