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, 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.

Derivation

Let $G=(V,E)$ be a graph of size$|V|=n$ and let $\lambda _1 \geq \lambda _2 \geq ... \geq \lambda _n$ be a non-increasing ordering of the eigenvalues of its adjacency matrix $A$ . The Estrada index is defined as

$EE(G)=\sum_{j=1}^n e^{\lambda _j}$

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

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

The subgraph centrality has the following closed form

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

where $\varphi _j (i)$ is the $i$ th entry of the $j$th eigenvector associated with the eigenvalue $\lambda _j$ . It is straightforward to realise that

$EE(G)=tr(e^A)$