Estrada index of cycles and paths

Let G be a graph on n vertices, and let λ1, λ2, … ,λn be its eigenvalues. The Estrada index of G is a recently introduced molecular structure descriptor, defined as EE ( G ) = ∑ i = 1 n e λ i . We show that the Estrada indices of the n-vertex cycle Cn and the n-vertex path Pn can be approximated as EE(Cn) ≈ n I0 and EE(Pn) ≈ (n + 1)I0 − cosh(2), where I 0 = ∑ k ⩾ 0 1 / ( k ! ) 2 = 2.27958530 … . The precision of these approximations is remarkably good.