NEW UPPER BOUND ON THE LARGEST LAPLACIAN EIGENVALUE OF GRAPHS

Let G = (V;E) be a simple, undirected graph with maximum and minimum degree ∆ and respectively, and let A be the adjacency matrix and Q be the Laplacianmatrix of G. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been applied to several elds, such as randomized algorithms, combinatorial optimization problems and machine learning. In this paper, we compute lower and upper bounds for the largest Laplacian eigenvalue which is related with a given maximum and minimum degree and a given number of vertices and edges. We also compare our results in this paper with some known results.