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 "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]

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