Spectral radii of graphs with given chromatic number

Abstract

We consider the set G n , k of graphs of order n with the chromatic number k ≥ 2 . In this note, we prove that in G n , k the Turán graph T n , k has the maximal spectral radius; and P n if k = 2 , C n if k = 3 and n is odd, C n − 1 1 if k = 3 and n is even, K k ( l ) if k ≥ 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.