The minimum spectral radius of graphs with a given clique number

Dragan Stevanovic, Pierre Hansen
2008 The Electronic Journal of Linear Algebra  
In this paper, it is shown that among connected graphs with maximum clique size ω, the minimum value of the spectral radius of adjacency matrix is attained for a kite graph P K n−ω,ω , which consists of a complete graph Kω to a vertex of which a path P n−ω is attached. For any fixed ω, a small interval to which the spectral radii of kites P Km,ω, m ≥ 1, belong is exhibited. From the proof of Lemma 2.1, it is evident that P S u (λ) > P S v (λ) for all λ > ρ(S v
more » ... ) for all λ > ρ(S v ). P k1 , P k2 , . . . , P km attached to m distinct vertices of K. With the repeated use of Lemma 2.1 to paths P k1 and P ki , 2 ≤ i ≤ m, we may decrease the spectral radius of G 2 until the attached paths P k2 , . . . , P km disappear, and we finally arrive to the kite graph P K n−ω,ω . Since we have (strictly) decreased the spectral radius at each step, we may conclude that the kite graph P K n−ω,ω has smaller spectral radius than any other graph in G n,ω . This finishes the proof of Theorem 1.1.
doi:10.13001/1081-3810.1253 fatcat:2txm6phfwfdepprgkrmi4rf2xy