Let G be a simple graph of order n with eigenvalues λ1, λ2, · · · , λn and normalized Laplacian eigenvalues μ1,μ2, · · · ,μn. The Estrada index and normalized Laplacian Estrada index are defined as EE(G) = ∑n k=1 e λk and LEE(G) = ∑n k=1 e μk−1, respectively. We establish upper and lower bounds to EE and LEE for edge-independent random graphs, containing the classical Erdös-Rényi graphs as spec...

The logarithm of the Estrada index has been recently proposed as a spectral measure to characterize the robustness of complex networks. We derive novel analytic lower bounds for the logarithm of the Estrada index based on the Laplacian spectrum and the mixing times of random walks on the network. The main techniques employed are some inequalities, such as the thermodynamic inequality in statist...

‎Let $Gamma$ be a graph with adjacency eigenvalues $lambda_1leqlambda_2leqldotsleqlambda_n$‎. ‎Then the energy of‎ ‎$Gamma$‎, ‎a concept defined in 1978 by Gutman‎, ‎is defined as $mathcal{E}(G)=sum_{i=1}^n|lambda_i|$‎. ‎Also‎ ‎the Estrada index of $Gamma$‎, ‎which is defined in 2000 by Ernesto Estrada‎, ‎is defined as $EE(Gamma)=sum_{i=1}^ne^{lambda_i}$‎. ‎In this paper‎, ‎we compute the eigen...

Abstract The Laplacian Estrada index of a graphG is defined as LEE(G) = ∑n i=1 e μi , where μ1 ≥ μ2 ≥ · · · ≥ μn−1 ≥ μn = 0 are the eigenvalues of its Laplacian matrix. An unsolved problem in [19] is whether Sn(3, n − 3) or Cn(n − 5) has the third maximal Laplacian Estrada index among all trees on n vertices, where Sn(3, n − 3) is the double tree formed by adding an edge between the centers of ...

In this talk I review three fatal flaws of the DGP braneworld model, which has been put forward as a possible model for late time acceleration without a cosmological constant; Ghosts, Cosmological Crashes, and Instability of the 5D vacuum. The talk is based on work in collaboration with Charmousis, Kaloper, Myers and Padilla. 2) (The title refers to a film: The three burials of Melquiades Estra...