Lower bounds for Estrada index and Laplacian Estrada index

Let GG be an nn-vertex graph. If λ1,λ2,…,λnλ1,λ2,…,λn and μ1,μ2,…,μnμ1,μ2,…,μn are the ordinary (adjacency) eigenvalues and the Laplacian eigenvalues of GG, respectively, then the Estrada index and the Laplacian Estrada index of GG are defined as EE(G)=∑i=1neλi and LEE(G)=∑i=1neμi, respectively. Some new lower bounds for EE and LEE are obtained and shown to be the best possible.