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 ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n$ be a non-increasing ordering of the eigenvalues of its adjacency matrix $A$. The Estrada index is defined as

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

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

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

The subgraph centrality has the following closed form

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

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

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

References

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