Abstract
The Estrada index of a graph/network is defined as the trace of the adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such as the Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many of these extensions, including new ones, such as Gaussian, Mittag–Leffler and Onsager ones. More importantly, we contextualize all of these indices in physico-mathematical frameworks which allow their interpretations and facilitate their extensions and further studies. We also describe several of the bounds and estimations of these indices reported in the literature and analyze many of them computationally for small graphs as well as large complex networks. This article is intended to formalize many of the Estrada indices proposed and studied in the mathematical literature serving as a guide for their further studies.
Highlights
At the dawn of the XXI century the current author proposed an index to quantify the “degree of folding” of a linear chain in a three-dimensional space [70]
We presented an account of the many different facets of the Estrada indices of graphs
The Seidel Estrada index is placed in the context of signed graphs, the theory of balance and the concept of network bipartivity
Summary
At the dawn of the XXI century the current author proposed an index to quantify the “degree of folding” of a linear chain in a three-dimensional space [70]. Estrada λ j (W ) are the eigenvalues of certain tridiagonal matrix W whose diagonal entries are related to the cosines of the dihedral angles between adjacent planes and Wi,i+1 and Wi+1,i are equal to one This index characterizes very well the degree of folding of a geometric chain and it has been mainly applied to the study of the degree of folding of proteins (see for instance [71,73,211]), it can be applied to the folding of any linear chain. For this purpose we include some numerical analysis of these bounds in the set of 11,117 connected graphs with 8 nodes and in five real-world networks representing a variety of complex system scenarios. The paper is intended as a guide for further studies and developments in this area of spectral graph theory
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