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

Let image be a simple connected weighted graph on n vertices in which the edge weights are positive numbers. Denote by i not, vert, similar j if the vertices i and j are adjacent and by wi,j the weight of the edge ij. Let image. Let λ1 be the largest Laplacian eigenvalue of image. We first derive the upper boundimageWe call this bound the trivial upper bound for λ1. Our main result isimageFor any image, this new bound does not exceed the trivial upper bound for λ1.