An improved upper bound for the Laplacian spectral radius of graphs

Let G be a simple graph with n vertices, m edges. Let Δ and δ be the maximum and minimum degree of G , respectively. If each edge of G belongs to t triangles ( t ≥ 1 ), then we present a new upper bound for the Laplacian spectral radius of G as follows: λ 1 ( G ) ≤ 2 Δ − t + ( 2 Δ − t ) 2 + 8 m − 4 δ ( n − 1 ) − 4 δ 2 + 4 ( δ − 1 ) Δ 2 . Moreover, we give an example to illustrate that our result is, in some cases, the best.