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 "Estrada index" was introduced by de la Peña et al. in 2007.

Derivation

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

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

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 $i$ is defined as

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

The subgraph centrality has the following closed form

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

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

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

References

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