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

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

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 ${\displaystyle i}$ is defined as[3]

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

The subgraph centrality has the following closed form[3]

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

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

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

## References

1. ^ Estrada, E. (2000). "Characterization of 3D molecular structure". Chem. Phys. Lett. 319 (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. ^ a b c 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. PMID 16089598. S2CID 4512786.