The Estrada Index: an Updated Survey

Ivan Gutman, Hanyuan Deng, Slavko Radenković

If $\lambda_i$, $i=1,2,\ldots,n$, are the eigenvalues of the graph $G$, then the Estrada index $EE$ of $G$ is the sum of the terms $e^{\lambda_i}$. This graph invariant appeared for the first time in year 2000, in a paper by Ernesto Estrada, dealing with the folding of protein molecules. Since then a remarkable variety of other chemical and non-chemical applications of $EE$ were communicated. The mathematical studies of the Estrada index started only a few years ago. Until now a number of lower and upper bounds were obtained, and the problem of extremal $EE$ for trees solved. Also, approximations and correlations for $EE$ were put forward, valid for chemically interesting molecular graphs. This chapter in an updated version of the an earlier survey by the same authors, published in the book D. Cvetković, I. Gutman (Eds.), \emph{Applications of Graph Spectra}, Math. Inst., Belgrade, 2009, pp. 123-140.