Note on Estrada and $L$-Estrada indices of graphs

G. H.,Fath-Tabar, A. R.,Ashrafi, I.,Gutman

Let $G$ be a graph of order $n$\,. Let $\lambda_1,\lambda_2,\ldots,\lambda_n$ be its eigenvalues and $\mu_1,\mu_2,\ldots,\mu_n$ its Laplacian eigenvalues. The Estrada index $EE$ of the graph $G$ is defined as the sum of the terms $e^{\lambda_i} \ , \ i=1,2,\ldots,n$\,. In this paper the notion of Laplacian--Estrada index ($L$-Estrada index, $LEE$) of a graph is introduced. It is defined as the sum of the terms $e^{\mu_i} \ , i=1,2,Ĺ‚dots,n$\,. The basic properties of $LEE$ are established, and compared with the analogous properties of $EE$\,. In addition, the Estrada and $L$-Estrada indices of some important classes of graphs are computed.