Spectral radius of digraphs with given dichromatic number

Let D be a digraph with vertex set V(D). A partition of V(D) into k acyclic sets is called a k-coloring of D. The minimum integer k for which there exists a k-coloring of D is the dichromatic number χ(D) of the digraph D. Denote Gn,k the set of the digraphs of order n with the dichromatic number k≥2. In this note, we characterize the digraph which has the maximal spectral radius in Gn,k. Our result generalizes the result of [8] by Feng et al.