Estrada index explained
In chemical graph theory, the Estrada index is a topological index of protein folding. The index was first defined by Ernesto Estrada as a measure of the degree of folding of a protein,[1] which is represented as a path-graph weighted by the dihedral or torsional angles of the protein backbone. This index of degree of folding has found multiple applications in the study of protein functions and protein-ligand interactions.
The name "Estrada index" was introduced by de la Peña et al. in 2007.[2]
Derivation
Let
be a graph of size
and let
be a non-increasing ordering of the eigenvalues of its
adjacency matrix
. The Estrada index is defined as
| n |
\operatorname{EE}(G)=\sum | |
| j=1 |
For a general graph, the index can be obtained as the sum of the subgraph centralities of all nodes in the graph. The subgraph centrality of node
is defined as
[3] | infty |
\operatorname{EE}(i)=\sum | |
| k=0 |
The subgraph centrality has the following closed form
[3] | A) |
\operatorname{EE}(i)=(e | |
| ii |
(i)]2
where
is the
th entry of the
th eigenvector associated with the eigenvalue
. It is straightforward to realise that
[3] \operatorname{EE}(G)=\operatorname{tr}(eA)
References
- Bo. Zhou. Ivan. Gutman. 10.2298/AADM0902371Z. More on the Laplacian Estrada Index. 2009. 3. 2. 371–378. Appl. Anal. Discrete Math.. free.
Notes and References
- E. . Estrada. Chem. Phys. Lett. . 319 . 713 . 2000. 10.1016/S0009-2614(00)00158-5. Characterization of 3D molecular structure . 319. 2000CPL...319..713E.
- J. A.. de la Peña. I.. Gutman. J. . Rada. Estimating the Estrada index. Linear Algebra Appl. . 427. 2007. 70–76. 10.1016/j.laa.2007.06.020. free.
- E. . Estrada . J.A. . Rodríguez-Velázquez . Phys. Rev. E . 71 . 5 . 056103 . 2005. 10.1103/PhysRevE.71.056103. Subgraph centrality in complex networks. 16089598 . cond-mat/0504730 . 2005PhRvE..71e6103E . 4512786 .