Geodesic
Sharp Upper Bounds on the Signless Laplacian Spectral Radius of Strongly Connected Digraphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 4, pp. 977-988.

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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) = d_1^+ + d_2^+, (d_1^+ d_2^+ ) if and only if G is a star digraph K_1,n-1, where d_1^+, d_2^+ 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^+∑_v_j ∈ N_i^+ 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.
Keywords: digraph, signless Laplacian spectral radius 
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Xi, Weige; Wang, Ligong. Sharp Upper Bounds on the Signless Laplacian Spectral Radius of Strongly Connected Digraphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 4, pp. 977-988. http://geodesic.mathdoc.fr/item/DMGT_2016_36_4_a14/

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