Estrada index - Alchetron, The Free Social 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.

Derivation

Let G = ( V , E ) be a graph of size | V | = n and let λ 1 ≥ λ 2 ≥ . . . ≥ λ n be a non-increasing ordering of the eigenvalues of its adjacency matrix A . The Estrada index is defined as

E E ( G ) = ∑ j = 1 n e λ 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

E E ( i ) = ∑ k = 0 ∞ ( A k ) i i k !

The subgraph centrality has the following closed form

E E ( i ) = ( e A ) i i = ∑ j = 1 n [ φ ( i ) ] 2 e λ j

where φ j ( i ) is the i th entry of the j th eigenvector associated with the eigenvalue λ j . It is straightforward to realise that

E E ( G ) = t r ( e A )

References

Estrada index Wikipedia

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