Estrada index

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 of this index as the “Estrada index” was proposed by de la Peña et al. in 2007.^[2]

Derivation

Let $G=(V,E)$ be a graph of size$|V|=n$ and let ${\displaystyle \lambda _{1}\geq \lambda _{2}\geq ...\geq \lambda _{n}}$ be a non-increasing ordering of the eigenvalues of its adjacency matrix $A$ . The Estrada index is defined as

${\displaystyle EE(G)=\sum _{j=1}^{n}e^{\lambda _{j}}}$

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 $i$ is defined as^[3]

${\displaystyle EE(i)=\sum _{k=0}^{\infty }{\frac {(A^{k})_{ii}}{k!}}}$

The subgraph centrality has the following closed form^[3]

${\displaystyle EE(i)=(e^{A})_{ii}=\sum _{j=1}^{n}[\varphi (i)]^{2}e^{\lambda _{j}}}$

where ${\displaystyle \varphi _{j}(i)}$ is the $i$ th entry of the $j$th eigenvector associated with the eigenvalue ${\displaystyle \lambda _{j}}$ . It is straightforward to realise that^[3]

${\displaystyle EE(G)=tr(e^{A})}$

References

1. Estrada, E. (2000). "Characterization of 3D molecular structure". Chem. Phys. Lett. (319): 713. Bibcode:2000CPL...319..713E. doi:10.1016/S0009-2614(00)00158-5.
2. de la Peña, J. A.; Gutman, I.; Rada, J. (2007). "Estimating the Estrada index". Linear Algebra Appl. 427: 70–76. doi:10.1016/j.laa.2007.06.020.
3. Estrada, E.; Rodríguez-Velázquez, J.A. (2005). "Subgraph centrality in complex networks". Phys. Rev. E. 71 (5): 056103. arXiv:cond-mat/0504730. Bibcode:2005PhRvE..71e6103E. doi:10.1103/PhysRevE.71.056103.

• Zhou, Bo; Gutman, Ivan (2009). "More on the Laplacian Estrada Index". Applic. Anal. Discr. Math. 3 (2): 371–378. doi:10.2298/AADM0902371Z.