Estrada index

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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,[1] 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.[2]

Derivation[edit]

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 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[3]

The subgraph centrality has the following closed form[3]

where is the th entry of the th eigenvector associated with the eigenvalue . It is straightforward to realise that[3]

References[edit]

  1. ^ Estrada, E. (2000). "Characterization of 3D molecular structure". Chem. Phys. Lett. (319): 713. Bibcode:2000CPL...319..713E. doi:10.1016/S0009-2614(00)00158-5. 
  2. ^ de la Peña, J. A.; Gutman, I.; Rada, J. (2007). "Estimating the Estrada index". Linear Algebra Appl. 427: 70–76. doi:10.1016/j.laa.2007.06.020. 
  3. ^ a b c Estrada, E.; Rodríguez-Velázquez, J.A. (2005). "Subgraph centrality in complex networks". Phys. Rev. E. 71 (5): 056103. Bibcode:2005PhRvE..71e6103E. arXiv:cond-mat/0504730Freely accessible. doi:10.1103/PhysRevE.71.056103. 
  • Zhou, Bo; Gutman, Ivan (2009). "More on the Laplacian Estrada Index". Applic. Anal. Discr. Math. 3 (2): 371–378. doi:10.2298/AADM0902371Z.