The spectral radius of edge chromatic critical graphs

A connected graph G   with maximum degree Δ and edge chromatic number χ′(G)=Δ+1χ′(G)=Δ+1 is called Δ-critical if χ′(G−e)=Δχ′(G−e)=Δ for every edge e of G  . In this paper, we consider two weaker versions of Vizing's conjecture, which concern the spectral radius ρ(G)ρ(G) and the signless Laplacian spectral radius μ(G)μ(G) of G  . We obtain some lower bounds for ρ(G)ρ(G) and μ(G)μ(G), and present some cases where the conjectures are true. Finally, several open problems are also proposed.