On upper bounds for Laplacian graph eigenvalues

In this paper, we obtain the following upper bound for the largest Laplacian graph eigenvalue λ(G)λ(G):λ(G)⩽maxd(u)d(u)+m(u)+d(v)d(v)+m(v)d(u)+d(v)-2∑w∈N(u)∩N(v)d(w)d(u)+d(v),where the maximum is taken over all pairs (u,v)∈E(G)(u,v)∈E(G). This is an improvement on Li and Zhang’s result with -2∑w∈N(u)∩N(v)d(w)d(u)+d(v) omitted. We also present another new upper bound for λ(G)λ(G):λ(G)⩽maxd(u)d(v)m(u)+d(v)d(u)m(v):(u,v)∈E(G).