Bounds for the Laplacian spectral radius of graphs

Abstract

Let G be a simple graph with n vertices, m edges, diameter D and degree sequence d 1, d 2, …, d n , and let λ1(G) be the largest Laplacian eigenvalue of G. Denote Δ = max{d i : 1 ≤ i ≤ n}, and , where α is a real number. In this article, we first give an upper bound on λ1(G) for a non-regular graph involving Δ and D; next present two upper bounds on λ1(G) for a connected graph in terms of d i and (α m) i ; at last obtain a lower bound on λ1(G) for a connected bipartite graph in terms of d i and (α t) i . Some known results are shown to be the consequences of our theorems.