A new upper bound for the Laplacian spectral radius of graphs

Abstract

Let G = ( V, E) be a graph on n vertices. Denote by d i = d( v i ) the degree of v i ∈ V( G). Then λ ( G ) ⩽ max d i + d i 2 + 8 d i m i ′ 2 , v i ∈ V ( G ) , where m i ′ = ∑ v i v j ∈ E ( d j - ∣ N i ∩ N j ∣ ) d i , ∣ N i ∩ N j ∣ is the number of common neighbors of v i and v j . Moreover, equality holds if and only if G is a bipartite regular graph.