Spectral radii of graphs with given chromatic number

We consider the set Gn,kGn,k of graphs of order nn with the chromatic number k≥2k≥2. In this note, we prove that in Gn,kGn,k the Turán graph Tn,kTn,k has the maximal spectral radius; and PnPn if k=2k=2, CnCn if k=3k=3 and nn is odd, Cn−11 if k=3k=3 and nn is even, Kk(l) if k≥4k≥4 has the minimal spectral radius. Thus we answer a problem raised by Cao [D.S. Cao, Index function of graphs, J. East China Norm. Univ. Sci. Ed. 4 (1987) 1–8 (in Chinese). MR89m:05084] and Hong [Y. Hong, Bounds of eigenvalues of graphs, Discrete Math. 123 (1993) 65–74] in the affirmative.