Estrada and L-Estrada Indices of Edge-Independent Random Graphs

Let \(G\) be a simple graph of order \(n\) with eigenvalues \(\lambda_1,\lambda_2,\cdots,\lambda_n\) and normalized Laplacian eigenvalues \(\mu_1,\mu_2,\cdots,\mu_n\). The Estrada index and normalized Laplacian Estrada index are defined as \(EE(G)=\sum_{k=1}^ne^{\lambda_k}\) and \(\mathcal{L}EE(G)=\sum_{k=1}^ne^{\mu_k-1}\), respectively. We establish upper and lower bounds to \(EE\) and \(\mathcal{L}EE\) for edge-independent random graphs, containing the classical Erdos-Renyi graphs as special cases.

[1]  Shang Yi-Lun.  Local Natural Connectivity in Complex Networks , 2011 .

[2]  Ivan Gutman,et al.  LOWER BOUNDS FOR ESTRADA INDEX , 2008 .

[3]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[4]  Ernesto Estrada.  Characterization of 3D molecular structure , 2000 .

[5]  J. A. Peña,et al.  Estimating the Estrada index , 2007 .

[6]  D. Cvetkovic,et al.  Spectra of graphs : theory and application , 1995 .

[7]  Matthias Dehmer,et al.  Information Theory of Networks , 2011, Symmetry.

[8]  Ivan Gutman,et al.  Lower bounds for Estrada index and Laplacian Estrada index , 2010, Appl. Math. Lett..

[9]  Wai Chee Shiu,et al.  The normalized Laplacian Estrada index of a graph , 2014 .

[10]  Angel Garrido.  Classifying Entropy Measures , 2011, Symmetry.

[11]  Yilun Shang,et al.  Laplacian Estrada and Normalized Laplacian Estrada Indices of Evolving Graphs , 2015, PloS one.

[12]  J. A. Rodríguez-Velázquez,et al.  Subgraph centrality in complex networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Yi-Zheng Fan,et al.  Estrada Index of Random Graphs , 2012 .

[14]  Yilun Shang.  Lower bounds for the Estrada index using mixing time and Laplacian spectrum , 2013 .

[15]  Linyuan Lu,et al.  Spectra of Edge-Independent Random Graphs , 2012, Electron. J. Comb..

[16]  Jianping Liu,et al.  Bounds of the Estrada index of graphs , 2010 .

[17]  J. A. Rodríguez-Velázquez,et al.  Spectral measures of bipartivity in complex networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  J. A. Rodríguez-Velázquez,et al.  Atomic branching in molecules , 2006 .

[19]  Ioana Dumitriu,et al.  Sparse regular random graphs: Spectral density and eigenvectors , 2009, 0910.5306.

[20]  Riccardo Basosi,et al.  Complex Networks and Symmetry I: A Review , 2010, Symmetry.

[21]  Toufik Mansour,et al.  Relating Estrada index with spectral radius , 2007 .

[22]  R. Carbó-Dorca.  Smooth function topological structure descriptors based on graph-spectra , 2008 .

[23]  Bo Zhou,et al.  ON ESTRADA INDEX , 2008 .

[24]  Ernesto Estrada,et al.  Characterization of the folding degree of proteins , 2002, Bioinform..

[25]  Béla Bollobás,et al.  Random Graphs , 1985 .

[26]  Toufik Mansour,et al.  Estrada index and Chebyshev polynomials , 2008 .

[27]  Guido Caldarelli,et al.  Networks - A Very Short Introduction , 2012, Very short introductions.