Sharp upper bounds on the signless Laplacian spectral radius of strongly connected digraphs

Let G = (V(G),E(G)) be a simple strongly connected digraph and q(G) be the signless Laplacian spectral radius of G. For any vertex v_i ∈ V(G), let d_i⁺ denote the outdegree of v_i, m_i⁺ denote the average 2-outdegree of v_i, and N_i⁺ denote the set of out-neighbors of v_i. In this paper, we prove that: (1) q(G) = d₁⁺ + d₂⁺, (d₁⁺ ≠ d₂⁺) if and only if G is a star digraph K^↔_1,n-1, where d₁⁺,d₂⁺ are the maximum and the second maximum outdegree, respectively (K^↔_1,n-1 is the digraph on n vertices obtained from a star graph K_1,n-1 by replacing each edge with a pair of oppositely directed arcs). (2) q(G) ≤ max {1/2 (d_i⁺ + √d_i⁺² + 8d_i⁺m_i⁺) : v_i ∈ V(G)} with equality if and only if G is a regular digraph. (3) q(G) ≤ max {1/2 (d_i⁺ + √d_i⁺² + 4/d_i⁺ Σ_vj∈Ni+ d_j⁺(d_j⁺ + m_j⁺)) : v_i ∈ V(G)}. Moreover, the equality holds if and only if G is a regular digraph or a bipartite semiregular digraph. (4) q(G) ≤ max {1/2 (d_i⁺ + 2d_j⁺ - 1 + √(d_i⁺ - 2d_j⁺ + 1)² + 4d_i⁺) : (v_j, v_i) ∈ E(G)}. If the equality holds, then G is a regular digraph or G ∈ Ω, where Ω is a class of digraphs defined in this paper.