Upper Bounds for the Laplacian Graph Eigenvalues

We first apply non-negative matrix theory to the matrix K=D+A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ₁(G) of G and the spectral radius ρ(K) of K. And then by using this relation we present two upper bounds for λ₁(G) and determine the extrernal graphs which achieve the upper bounds. Supported by National Natural Science Foundation of China (Grant No. 19971086)