Estrada and \(\mathcal L\)-Estrada indices of edge-independent random graphs (Q2406242)

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