An improved upper bound for the Laplacian spectral radius of graphs

Let GG be a simple graph with nn vertices, mm edges. Let ΔΔ and δδ be the maximum and minimum degree of GG, respectively. If each edge of GG belongs to tt triangles (t≥1t≥1), then we present a new upper bound for the Laplacian spectral radius of GG as follows: λ1(G)≤2Δ−t+(2Δ−t)2+8m−4δ(n−1)−4δ2+4(δ−1)Δ2. Moreover, we give an example to illustrate that our result is, in some cases, the best.