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Estrada index
Topological index of protein folding

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 "Estrada index" was introduced by de la Peña et al. in 2007.

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Derivation

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

EE ⁡ ( G ) = ∑ j = 1 n e λ j {\displaystyle \operatorname {EE} (G)=\sum _{j=1}^{n}e^{\lambda _{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 {\displaystyle i} is defined as3

EE ⁡ ( i ) = ∑ k = 0 ∞ ( A k ) i i k ! {\displaystyle \operatorname {EE} (i)=\sum _{k=0}^{\infty }{\frac {(A^{k})_{ii}}{k!}}}

The subgraph centrality has the following closed form4

EE ⁡ ( i ) = ( e A ) i i = ∑ j = 1 n [ φ j ( i ) ] 2 e λ j {\displaystyle \operatorname {EE} (i)=(e^{A})_{ii}=\sum _{j=1}^{n}[\varphi _{j}(i)]^{2}e^{\lambda _{j}}}

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 that5

EE ⁡ ( G ) = tr ⁡ ( e A ) {\displaystyle \operatorname {EE} (G)=\operatorname {tr} (e^{A})}

References

  1. Estrada, E. (2000). "Characterization of 3D molecular structure". Chem. Phys. Lett. 319 (319): 713. Bibcode:2000CPL...319..713E. doi:10.1016/S0009-2614(00)00158-5. /wiki/Bibcode_(identifier) ↩

  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. https://doi.org/10.1016%2Fj.laa.2007.06.020 ↩

  3. Estrada, E.; Rodríguez-Velázquez, J.A. (2005). "Subgraph centrality in complex networks". Phys. Rev. E. 71 (5): 056103. arXiv:cond-mat/0504730. Bibcode:2005PhRvE..71e6103E. doi:10.1103/PhysRevE.71.056103. PMID 16089598. S2CID 4512786. /wiki/ArXiv_(identifier) ↩

  4. Estrada, E.; Rodríguez-Velázquez, J.A. (2005). "Subgraph centrality in complex networks". Phys. Rev. E. 71 (5): 056103. arXiv:cond-mat/0504730. Bibcode:2005PhRvE..71e6103E. doi:10.1103/PhysRevE.71.056103. PMID 16089598. S2CID 4512786. /wiki/ArXiv_(identifier) ↩

  5. Estrada, E.; Rodríguez-Velázquez, J.A. (2005). "Subgraph centrality in complex networks". Phys. Rev. E. 71 (5): 056103. arXiv:cond-mat/0504730. Bibcode:2005PhRvE..71e6103E. doi:10.1103/PhysRevE.71.056103. PMID 16089598. S2CID 4512786. /wiki/ArXiv_(identifier) ↩