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. 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. Let G = ( V , E ) {displaystyle G=(V,E)} be a graph of size | V | = n {displaystyle |V|=n} and let λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n {displaystyle lambda _{1}geq lambda _{2}geq cdots geq lambda _{n}} be a non-increasing ordering of the eigenvalues of its adjacency matrix A {displaystyle A} . 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 i {displaystyle i} is defined as The subgraph centrality has the following closed form where φ j ( i ) {displaystyle varphi _{j}(i)} is the i {displaystyle i} th entry of the j {displaystyle j} th eigenvector associated with the eigenvalue λ j {displaystyle lambda _{j}} . It is straightforward to realise that