Lower bounds for Estrada index and Laplacian Estrada index

Let G be an n -vertex graph. If λ 1 , λ 2 , … , λ n and μ 1 , μ 2 , … , μ n are the ordinary (adjacency) eigenvalues and the Laplacian eigenvalues of G , respectively, then the Estrada index and the Laplacian Estrada index of G are defined as EE ( G ) = ∑ i = 1 n e λ i and LEE ( G ) = ∑ i = 1 n e μ i , respectively. Some new lower bounds for EE and LEE are obtained and shown to be the best possible.