Bounds for the Laplacian spectral radius of graphs

Let G be a simple graph with n vertices, m edges, diameter D and degree sequence d(1), d(2), ..., d(n), and let lambda(1)(G) be the largest Laplacian eigenvalue of G. Denote Delta = max{d(i) : 1 <= i <= n}, ((alpha)t)(i) = Sigma(i similar to j) d(j)(alpha) and ((alpha)m)(i) = ((alpha)t)(i)/d(i)(alpha), where alpha is a real number. In this article, we first give an upper bound on lambda(1)(G) for a non-regular graph involving Delta and D; next present two upper bounds on lambda(1)(G) for a connected graph in terms of d(i) and ((alpha)m)(i); at last obtain a lower bound on lambda(1)(G) for a connected bipartite graph in terms of d(i) and ((alpha)t)(i). Some known results are shown to be the consequences of our theorems.