Signless Laplacian spectral radii of graphs with given chromatic number

Let G be a simple graph with vertices v(1), v(2), ... , v(n), of degrees Delta = d(1) >= d(2) >= ... d(n) = delta, respectively. LetA be the (0, 1)-adjacency matrix of G and D be the diagonal matrix diag(d(1), d(2), ... , d(n)). Q(G) = D + A is called the signless Laplacian of G. The largest eigenvalue of Q(G) is called the signless Laplacian spectral radius or Q-spectral radius of G. Denote by chi(G) the chromatic number for a graph G. In this paper, for graphs with order n, the extremal graphs with both the given chromatic number and the maximal Q-spectral radius are characterized, the extremal graphs with both the given chromatic number chi not equal 0 4, 5, 6, 7 and the minimal Q-spectral radius are characterized as well. (C) 2011 Elsevier Inc. All rights reserved.