Geodesic
The Minimum Spectral Radius of Signless Laplacian of Graphs with a Given Clique Number
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 1, pp. 95-102

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In this paper we observe that the minimal signless Laplacian spectral radius is obtained uniquely at the kite graph PK_n-ω,ω among all connected graphs with n vertices and clique number ω. In addition, we show that the spectral radius μ of PK_m,ω (m≥1) satisfies 1/2(2ω-1+√(4ω^2-12ω+17))≤μ≤ 2ω-1. More precisely, for m gt;1, μ satisfies the equation μ-ω-ω-1/μ-2ω+3=a_m√(μ^2-4μ)+1/t_1, where a_m=1/1-t_1^2m+3 and t_1=μ-2+√((μ-2)^2-4)/2. At last the spectral radius μ(PK_∞,ω) of the infinite graph PK_∞,ω is also discussed.
Keywords: clique number, kite graph, signless Laplacian, spectral radius 
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Su, Li; Li, Hong-Hai; Zhang, Jing. The Minimum Spectral Radius of Signless Laplacian of Graphs with a Given Clique Number. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 1, pp. 95-102. http://geodesic.mathdoc.fr/item/DMGT_2014_34_1_a7/