The Signless Laplacian Spectral Characterization of Strongly Connected Bicyclic Digraphs

Let −→ G be a digraph and A( −→ G) be the adjacency matrix of −→ G . Let D( −→ G) be the diagonal matrix with outdegrees of vertices of −→ G and Q( −→ G) = D( −→ G) + A( −→ G) be the signless Laplacian matrix of −→ G . The spectral radius of Q( −→ G) is called the signless Laplacian spectral radius of −→ G . In this paper, we determine the unique digraph which attains the maximum (or minimum) signless Laplacian spectral radius among all strongly connected bicyclic digraphs. Furthermore, we prove that any strongly connected bicyclic digraph is determined by the signless Laplacian spectrum.