The Estrada Index

Hanyuan Deng, Slavko Radenković, Ivan Gutman

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. In this paper the relevant results on the Estrada index are surveyed.