Characterization on graphs which achieve a Das' upper bound for Laplacian spectral radius

Let G = (V, E) be a graph on vertex set V = {v1,v2, … ,vn}. For any vertex vi, we denote by N(vi) the set of the vertices adjacent to vi in G. Das got the following upper bound for Laplacian spectral radius:λ1(G)⩽max{|N(vi)∪N(vj)|:1⩽i