Bounds for the eccentricity spectral radius of join digraphs with a fixed dichromatic number

The eccentricity matrix ɛ(G) of a strongly connected digraph G is defined as ɛ(G)_ij=d(v_i,v_j),ifd(v_i,v_j)=min{e⁺(v_i),e⁻(v_j)},0,otherwise.,where e⁺(v_i)=max{d(v_i,v_j)∣v_j∈V(G)} is the out-eccentricity of the vertex v_i of G, and e⁻(v_j)=max{d(v_i,v_j)∣v_i∈V(G)} is the in-eccentricity of the vertex v_j of G. The eigenvalue of ɛ(G) with the largest modulus is called the eccentricity spectral radius of G. In this paper, we obtain lower bounds for the eccentricity spectral radius among all join digraphs with a fixed dichromatic number. We also give upper bounds for the eccentricity spectral radius of some special join digraphs with a fixed dichromatic number.