chemical graph theory Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena. The pioneers of chemical graph theory are Alexandru Balaban, Ante Graovac, Iván Gutman, Haruo Hosoy ...

, the Estrada index is a

topological index In the fields of chemical graph theory, molecular topology, and mathematical chemistry, a topological index, also known as a connectivity index, is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compo ...

protein folding Protein folding is the physical process by which a protein chain is translated to its native three-dimensional structure, typically a "folded" conformation by which the protein becomes biologically functional. Via an expeditious and reproduci ...

. 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 angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...

s 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 In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...

A

. The Estrada index is defined as :

\operatorname(G)=\sum_^n e^

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(i)=\sum_^\infty \frac

The subgraph centrality has the following closed form :

\operatorname(i)=(e^A)_=\sum_^n varphi _j (i) 2 e^

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(G)=\operatorname(e^A)

References

*{{cite journal , first1=Bo , last1=Zhou , first2=Ivan , last2=Gutman , doi=10.2298/AADM0902371Z , title=More on the Laplacian Estrada Index , year=2009 , volume=3 , number=2 , pages=371–378 , journal=Appl. Anal. Discrete Math. , doi-access=free Mathematical chemistry Cheminformatics Graph invariants