Chromatic number and signless Laplacian spectral radius of graphs

For any simple graph G, the signless Laplacian matrix of G is defined as D(G)+A(G), where D(G) and A(G) are the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. %Let \chi(G) be the chromatic number of G Let q(G) be the signless Laplacian spectral radius of G (the largest eigenvalue of the signless Laplacian matrix of G). In this paper we find some relations between the chromatic number and the signless Laplacian spectral radius of graphs. In particular, we characterize all graphs G of order n with odd chromatic number \chi such that q(G)=۲n\Big(۱-\frac{۱}{\chi}\Big). Finally we show that if G is a graph of order n and with chromatic number \chi, then under certain conditions, q(G)<۲n\Big(۱-\frac{۱}{\chi}\Big)-\frac{۲}{n}. This result improves some previous similar results.

Keywords:

chromatic number , Majorization , Signless Laplacian matrix , Signless Laplacian spectral radius