547:
407:
186:
285:
713:
586:
653:
91:
127:
626:
606:
308:
209:
418:
316:
290:
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
132:
217:
661:
901:
40:
of the protein backbone. This index of degree of folding has found multiple applications in the study of protein functions and
33:
36:
as a measure of the degree of folding of a protein, which is represented as a path-graph weighted by the dihedral or
911:
41:
906:
17:
824:
741:
555:
631:
848:
814:
840:
25:
58:
878:
832:
780:
749:
189:
96:
805:
Estrada, E.; Rodríguez-Velázquez, J.A. (2005). "Subgraph centrality in complex networks".
37:
29:
828:
745:
611:
591:
542:{\displaystyle \operatorname {EE} (i)=(e^{A})_{ii}=\sum _{j=1}^{n}^{2}e^{\lambda _{j}}}
293:
194:
753:
402:{\displaystyle \operatorname {EE} (i)=\sum _{k=0}^{\infty }{\frac {(A^{k})_{ii}}{k!}}}
895:
852:
836:
785:
768:
844:
181:{\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}}
883:
866:
819:
280:{\displaystyle \operatorname {EE} (G)=\sum _{j=1}^{n}e^{\lambda _{j}}}
47:
The name "Estrada index" was introduced by de la Peña et al. in 2007.
732:
Estrada, E. (2000). "Characterization of 3D molecular structure".
708:{\displaystyle \operatorname {EE} (G)=\operatorname {tr} (e^{A})}
188:
be a non-increasing ordering of the eigenvalues of its
412:
The subgraph centrality has the following closed form
664:
634:
614:
594:
558:
421:
319:
296:
220:
197:
135:
99:
61:
707:
647:
620:
600:
580:
541:
401:
302:
279:
203:
180:
121:
85:
767:de la Peña, J. A.; Gutman, I.; Rada, J. (2007).
628:th eigenvector associated with the eigenvalue
8:
882:
818:
800:
798:
796:
784:
696:
663:
639:
633:
613:
593:
563:
557:
531:
526:
516:
497:
484:
473:
457:
447:
420:
379:
369:
359:
353:
342:
318:
295:
269:
264:
254:
243:
219:
196:
172:
153:
140:
134:
108:
100:
98:
60:
655:. It is straightforward to realise that
724:
867:"More on the Laplacian Estrada Index"
7:
354:
211:. The Estrada index is defined as
14:
32:. The index was first defined by
865:Zhou, Bo; Gutman, Ivan (2009).
581:{\displaystyle \varphi _{j}(i)}
769:"Estimating the Estrada index"
702:
689:
677:
671:
575:
569:
513:
509:
503:
490:
454:
440:
434:
428:
376:
362:
332:
326:
233:
227:
109:
101:
80:
68:
1:
754:10.1016/S0009-2614(00)00158-5
648:{\displaystyle \lambda _{j}}
42:protein-ligand interactions
928:
837:10.1103/PhysRevE.71.056103
871:Appl. Anal. Discrete Math
786:10.1016/j.laa.2007.06.020
86:{\displaystyle G=(V,E)}
902:Mathematical chemistry
709:
649:
622:
602:
582:
543:
489:
403:
358:
304:
281:
259:
205:
182:
123:
87:
710:
650:
623:
603:
583:
544:
469:
404:
338:
305:
282:
239:
206:
183:
124:
122:{\displaystyle |V|=n}
88:
18:chemical graph theory
884:10.2298/AADM0902371Z
662:
632:
612:
592:
556:
419:
317:
294:
218:
195:
133:
97:
59:
829:2005PhRvE..71e6103E
773:Linear Algebra Appl
746:2000CPL...319..713E
93:be a graph of size
705:
645:
618:
598:
578:
539:
399:
300:
277:
201:
178:
119:
83:
621:{\displaystyle j}
608:th entry of the
601:{\displaystyle i}
397:
303:{\displaystyle i}
204:{\displaystyle A}
26:topological index
919:
912:Graph invariants
888:
886:
857:
856:
822:
820:cond-mat/0504730
802:
791:
790:
788:
764:
758:
757:
734:Chem. Phys. Lett
729:
714:
712:
711:
706:
701:
700:
654:
652:
651:
646:
644:
643:
627:
625:
624:
619:
607:
605:
604:
599:
587:
585:
584:
579:
568:
567:
548:
546:
545:
540:
538:
537:
536:
535:
521:
520:
502:
501:
488:
483:
465:
464:
452:
451:
408:
406:
405:
400:
398:
396:
388:
387:
386:
374:
373:
360:
357:
352:
309:
307:
306:
301:
286:
284:
283:
278:
276:
275:
274:
273:
258:
253:
210:
208:
207:
202:
190:adjacency matrix
187:
185:
184:
179:
177:
176:
158:
157:
145:
144:
128:
126:
125:
120:
112:
104:
92:
90:
89:
84:
38:torsional angles
927:
926:
922:
921:
920:
918:
917:
916:
907:Cheminformatics
892:
891:
864:
861:
860:
804:
803:
794:
766:
765:
761:
731:
730:
726:
721:
692:
660:
659:
635:
630:
629:
610:
609:
590:
589:
559:
554:
553:
527:
522:
512:
493:
453:
443:
417:
416:
389:
375:
365:
361:
315:
314:
292:
291:
265:
260:
216:
215:
193:
192:
168:
149:
136:
131:
130:
95:
94:
57:
56:
53:
34:Ernesto Estrada
30:protein folding
12:
11:
5:
925:
923:
915:
914:
909:
904:
894:
893:
890:
889:
877:(2): 371–378.
859:
858:
792:
759:
723:
722:
720:
717:
716:
715:
704:
699:
695:
691:
688:
685:
682:
679:
676:
673:
670:
667:
642:
638:
617:
597:
577:
574:
571:
566:
562:
550:
549:
534:
530:
525:
519:
515:
511:
508:
505:
500:
496:
492:
487:
482:
479:
476:
472:
468:
463:
460:
456:
450:
446:
442:
439:
436:
433:
430:
427:
424:
410:
409:
395:
392:
385:
382:
378:
372:
368:
364:
356:
351:
348:
345:
341:
337:
334:
331:
328:
325:
322:
310:is defined as
299:
288:
287:
272:
268:
263:
257:
252:
249:
246:
242:
238:
235:
232:
229:
226:
223:
200:
175:
171:
167:
164:
161:
156:
152:
148:
143:
139:
118:
115:
111:
107:
103:
82:
79:
76:
73:
70:
67:
64:
52:
49:
13:
10:
9:
6:
4:
3:
2:
924:
913:
910:
908:
905:
903:
900:
899:
897:
885:
880:
876:
872:
868:
863:
862:
854:
850:
846:
842:
838:
834:
830:
826:
821:
816:
813:(5): 056103.
812:
808:
801:
799:
797:
793:
787:
782:
778:
774:
770:
763:
760:
755:
751:
747:
743:
739:
735:
728:
725:
718:
697:
693:
686:
683:
680:
674:
668:
665:
658:
657:
656:
640:
636:
615:
595:
572:
564:
560:
532:
528:
523:
517:
506:
498:
494:
485:
480:
477:
474:
470:
466:
461:
458:
448:
444:
437:
431:
425:
422:
415:
414:
413:
393:
390:
383:
380:
370:
366:
349:
346:
343:
339:
335:
329:
323:
320:
313:
312:
311:
297:
270:
266:
261:
255:
250:
247:
244:
240:
236:
230:
224:
221:
214:
213:
212:
198:
191:
173:
169:
165:
162:
159:
154:
150:
146:
141:
137:
116:
113:
105:
77:
74:
71:
65:
62:
50:
48:
45:
43:
39:
35:
31:
27:
23:
22:Estrada index
19:
874:
870:
810:
807:Phys. Rev. E
806:
776:
772:
762:
740:(319): 713.
737:
733:
727:
551:
411:
289:
54:
46:
21:
15:
896:Categories
719:References
51:Derivation
779:: 70–76.
687:
669:
637:λ
561:φ
529:λ
495:φ
471:∑
426:
355:∞
340:∑
324:
267:λ
241:∑
225:
170:λ
166:≥
163:⋯
160:≥
151:λ
147:≥
138:λ
845:16089598
129:and let
853:4512786
825:Bibcode
742:Bibcode
588:is the
552:where
851:
843:
20:, the
849:S2CID
815:arXiv
24:is a
841:PMID
55:Let
879:doi
833:doi
781:doi
777:427
750:doi
738:319
28:of
16:In
898::
873:.
869:.
847:.
839:.
831:.
823:.
811:71
809:.
795:^
775:.
771:.
748:.
736:.
684:tr
666:EE
423:EE
321:EE
222:EE
44:.
887:.
881::
875:3
855:.
835::
827::
817::
789:.
783::
756:.
752::
744::
703:)
698:A
694:e
690:(
681:=
678:)
675:G
672:(
641:j
616:j
596:i
576:)
573:i
570:(
565:j
533:j
524:e
518:2
514:]
510:)
507:i
504:(
499:j
491:[
486:n
481:1
478:=
475:j
467:=
462:i
459:i
455:)
449:A
445:e
441:(
438:=
435:)
432:i
429:(
394:!
391:k
384:i
381:i
377:)
371:k
367:A
363:(
350:0
347:=
344:k
336:=
333:)
330:i
327:(
298:i
271:j
262:e
256:n
251:1
248:=
245:j
237:=
234:)
231:G
228:(
199:A
174:n
155:2
142:1
117:n
114:=
110:|
106:V
102:|
81:)
78:E
75:,
72:V
69:(
66:=
63:G
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