Estrada index
Encyclopedia
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 and many of its mathematical properties are now available in the literature.
be a graph of size
and let 
be a non-increasing ordering of the eigenvalues of its adjacency matrix 
. The Estrada index is defined as
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
is defined as
The subgraph centrality has the following closed form
where
is the 
th entry of the 
th eigenvector associated with the eigenvalue 
. It is straightforward to realise that
Chemical graph theory
Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena....
, the Estrada index is a topological index
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 compound. Topological indices are numerical parameters...
of protein folding
Protein folding
Protein folding is the process by which a protein structure assumes its functional shape or conformation. It is the physical process by which a polypeptide folds into its characteristic and functional three-dimensional structure from random coil....
. The index was first defined by Ernesto Estrada
Ernesto Estrada
Ernesto Estrada is a Cuban-Spanish scientist. He is the Chair in Complexity Science, a full Professor at the Department of Physics and Department of Mathematics, and a member of the Institute of Complex Systems of the University of Strathclyde, Glasgow, United Kingdom...
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 and many of its mathematical properties are now available in the literature.
Derivation
Let be a graph of size and let be a non-increasing ordering of the eigenvalues of its adjacency matrix . The Estrada index is defined asFor 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
is defined asThe subgraph centrality has the following closed form
where
is the th entry of the th eigenvector associated with the eigenvalue . It is straightforward to realise that