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 [math]\displaystyle{ G=(V,E) }[/math] be a graph of size [math]\displaystyle{ |V|=n }[/math] and let [math]\displaystyle{ \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n }[/math] be a non-increasing ordering of the eigenvalues of its adjacency matrix [math]\displaystyle{ A }[/math]. The Estrada index is defined as

[math]\displaystyle{ \operatorname{EE}(G)=\sum_{j=1}^n e^{\lambda_j} }[/math]

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 [math]\displaystyle{ i }[/math] is defined as

[math]\displaystyle{ \operatorname{EE}(i)=\sum_{k=0}^\infty \frac{(A^k)_{ii}} {k!} }[/math]

The subgraph centrality has the following closed form

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

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

[math]\displaystyle{ \operatorname{EE}(G)=\operatorname{tr}(e^A) }[/math]

References

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