A nontrivial upper bound on the largest Laplacian eigenvalue of weighted graphs

Let GG be a simple connected weighted graph on n vertices in which the edge weights are positive numbers. Denote by i ∼ j if the vertices i and j are adjacent and by wi,j the weight of the edge ij  . Let wi=∑j=1nwi,j. Let λ1 be the largest Laplacian eigenvalue of GG. We first derive the upper boundλ1⩽∑j=1nmaxk∼jwk,j.We call this bound the trivial upper bound for λ1. Our main result isλ1⩽12maxi∼jwi+wj+∑k∼i,k≁jwi,k+∑k∼j,k≁iwj,k+∑k∼i,k∼j|wi,k-wj,k|.For any GG, this new bound does not exceed the trivial upper bound for λ1.