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 vi ∈ V (G), let d+i denote the outdegree of vi, m+i denote the average 2-outdegree of vi, and N+i denote the set of out-neighbors of vi. In this paper, we prove that: (1) (1) q(G) = d+1 +d+2 , (d+1 ≠ d+2) if and only if G is a star digraph [...] ,where d+1, d+2 are the maximum and the second maximum outdegree, respectively [...] is the digraph on n vertices obtained from a star graph K1,n−1 by replacing each edge with a pair of oppositely directed arcs). (2) [...] with equality if and only if G is a regular digraph. (3) [...] Moreover, the equality holds if and only if G is a regular digraph or a bipartite semiregular digraph. (4) [...] . If the equality holds, then G is a regular digraph or G ∈Ω, where is a class of digraphs defined in this paper.
@article{bwmeta1.element.doi-10_7151_dmgt_1915,
author = {Weige Xi and Ligong Wang},
title = {Sharp Upper Bounds on the Signless Laplacian Spectral Radius of Strongly Connected Digraphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {36},
year = {2016},
pages = {977-988},
zbl = {1350.05099},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1915}
}
Weige Xi; Ligong Wang. Sharp Upper Bounds on the Signless Laplacian Spectral Radius of Strongly Connected Digraphs. Discussiones Mathematicae Graph Theory, Tome 36 (2016) pp. 977-988. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1915/