The Signless Laplacian Spectral Radius of Some Strongly Connected Digraphs

Let G→ be a strongly connected digraph and Q(G→) be the signless Laplacian matrix of G→. The spectral radius of Q(G→) is called the signless Lapliacian spectral radius of G→. Let ∞˜ ₁-digraph and ∞˜ ₂-digraph be two kinds of generalized strongly connected 1-digraphs and let θ˜ ₁-digraph and θ˜ ₂-digraph be two kinds of generalized strongly connected µ-digraphs. In this paper, we determine the unique digraph which attains the maximum(or minimum) signless Laplacian spectral radius among all ∞˜ ₁-digraphs and θ˜ ₁-digraphs. Furthermore, we characterize the extremal digraph which achieves the maximum signless Laplacian spectral radius among ∞˜ ₂-digraphs and θ˜ ₂-digraphs, respectively.