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

The eccentricity matrix epsilon ( G ) of a strongly connected digraph G is defined as { d ( v i , v j ) , if d ( v i , v j ) = min { e + ( v i ) , e - ( v j ) } , epsilon ( G ) ij = 0 , otherwise ., where e + ( v i ) = max { d ( v i , v j ) | v j E 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 E V( G ) } is the in -eccentricity of the vertex v j of G . The eigenvalue of epsilon ( 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. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).