Estimating the Estrada index

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Quadrature rule-based bounds for functions of adjacency matrices (2010, background): “Recent work has often focused on the definition and evaluation of computable quantities that describe interesting characteristics of a given network or of its parts.”
Approximating the Spectral Sums of Large-scale Matrices using Chebyshev Approximations (2016, background): “The Estrada index (Σexp(A)) has been initially developed for topological index of protein folding in the study of protein functions and protein-ligand interactions [17, 14], and currently it appears in numerous other applications, e.g., statistical thermodynamics [20, 19], information theory [9]…”
On the Harary Energy and Harary Estrada Index of a Graph (2010, unknown): “Since dij ≥ 1 for i 6= j and there are n(n − 1)/2 pairs of vertices in G, by the upper bound of Theorem 2.1, we get HE(G) ≤ √√√√2n ∑ 1≤i j≤n ( 1 dij )2 ≤ √ 2n n(n− 1) 2 = n √ n− 1.”