Estrada index

<h2 id="derivation">Derivation</h2>
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 <a href="/facts/Adjacency_matrix/jYXNhYGs">adjacency matrix</a> 
 
 
 
 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 as<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

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 form<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

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 that<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

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

<ul><li>Zhou, Bo; Gutman, Ivan (2009). <a href="https://doi.org/10.2298%2FAADM0902371Z">"More on the Laplacian Estrada Index"</a>. Appl. Anal. Discrete Math. 3 (2): 371–378. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2298%2FAADM0902371Z">10.2298/AADM0902371Z</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="https://doi.org/10.1016%2Fj.laa.2007.06.020" target="_blank">https://doi.org/10.1016%2Fj.laa.2007.06.020</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

Estrada index open-in-new

Estrada index