THE MINIMUM SPECTRAL RADIUS OF SIGNLESS LAPLACIAN OF GRAPHS WITH A GIVEN CLIQUE NUMBER

In this paper we observe that the minimal signless Laplacian spectral radius is obtained uniquely at the kite graph PKn-omega omega among all connected graphs with n vertices and clique number w. In addition, we show that the spectral radius mu of PKm,omega (m >= 1) satisfies 1/2(2 omega -1 + root 4 omega(2) - 12 omega + 17) <= mu <= 2 omega - 1. More precisely, for m > 1, mu satisfies the equation mu - omega - omega-1/mu-2 omega + 3 = a(m) root mu(2) - 4 mu + 1/l(1), where a(m) = 1/1=t(1)(2m) (broken vertical bar) (3a) and t(1) = mu-2+root(mu-2)(2-4). At last the spectral radius mu(PK infinity,omega) of the infinite graph PK is also discussed.