The many facets of the Estrada indices of graphs and networks

Abstract The Estrada index of a graph/network is defined as the trace adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many these extensions, including new ones, Gaussian, Mittag–Leffler and Onsager ones. More importantly, contextualize all indices in physico-mathematical frameworks which allow their interpretations facilitate extensions further studies. We also several bounds estimations reported literature analyze them computationally for small graphs well large complex networks. This article intended formalize proposed studied mathematical serving guide