A new upper bound for the Laplacian spectral radius of graphs

Let G = (V, E) be a graph on n vertices. Denote by di = d(vi) the degree of vi ∈ V(G). Thenλ(G)⩽maxdi+di2+8dimi′2,vi∈V(G),where mi′=∑vivj∈E(dj-∣Ni∩Nj∣)di, ∣Ni ∩ Nj∣ is the number of common neighbors of vi and vj. Moreover, equality holds if and only if G is a bipartite regular graph.