1738:
633:
410:
417:
320:
628:{\displaystyle {\begin{pmatrix}4&0&0&0&0&0\\0&3&0&0&0&0\\0&0&2&0&0&0\\0&0&0&3&0&0\\0&0&0&0&3&0\\0&0&0&0&0&1\\\end{pmatrix}}}
215:
362:
1396:
204:
99:
640:
Note that in the case of undirected graphs, an edge that starts and ends in the same node increases the corresponding degree value by 2 (i.e. it is counted twice).
135:
670:
178:
158:
1610:
829:
1701:
1620:
1386:
787:
1774:
1421:
968:
315:{\displaystyle D_{i,j}:=\left\{{\begin{matrix}\deg(v_{i})&{\mbox{if}}\ i=j\\0&{\mbox{otherwise}}\end{matrix}}\right.}
782:, Encyclopedia of Mathematics and its Applications, vol. 102, Cambridge University Press, Cambridge, pp. 113–136,
1185:
822:
1260:
1416:
938:
1520:
1391:
1305:
1625:
1515:
1223:
903:
1660:
1589:
1471:
1331:
928:
815:
1530:
1113:
918:
680:
24:
1476:
1213:
1063:
1058:
893:
868:
863:
44:
40:
1737:
1670:
1028:
858:
838:
719:
398:
1691:
1665:
1243:
1048:
1038:
55:
of a graph: the
Laplacian matrix is the difference of the degree matrix and the adjacency matrix.
1742:
1696:
1686:
1640:
1635:
1564:
1500:
1366:
1103:
1098:
1033:
1023:
888:
676:
328:
183:
1779:
1753:
1540:
1535:
1525:
1505:
1466:
1461:
1290:
1285:
1270:
1265:
1256:
1251:
1198:
1093:
1043:
988:
958:
953:
933:
923:
883:
783:
777:
755:
66:
1748:
1716:
1645:
1584:
1579:
1559:
1495:
1401:
1371:
1356:
1341:
1336:
1275:
1228:
1203:
1193:
1164:
1083:
1078:
1053:
983:
963:
873:
853:
745:
727:
365:
52:
48:
32:
797:
741:
104:
1446:
1381:
1361:
1346:
1326:
1310:
1208:
1139:
1129:
1088:
973:
943:
793:
737:
649:
206:
36:
723:
1706:
1650:
1630:
1615:
1574:
1451:
1411:
1376:
1300:
1239:
1218:
1159:
1149:
1134:
1068:
1013:
1003:
998:
908:
655:
369:
163:
143:
750:
1768:
1711:
1510:
1441:
1431:
1426:
1351:
1280:
1154:
1144:
1073:
993:
978:
913:
47:—that is, the number of edges attached to each vertex. It is used together with the
1594:
1551:
1456:
1169:
1108:
1018:
898:
1436:
1406:
1174:
1008:
878:
773:
364:
of a vertex counts the number of times an edge terminates at that vertex. In an
20:
712:
Proceedings of the
National Academy of Sciences of the United States of America
1487:
948:
1721:
1295:
776:(2004), "Graph Laplacians", in Beineke, Lowell W.; Wilson, Robin J. (eds.),
732:
703:
381:
759:
683:
of the degree matrix is twice the number of edges of the considered graph.
368:, this means that each loop increases the degree of a vertex by two. In a
1655:
377:
409:
707:
392:
The following undirected graph has a 6x6 degree matrix with values:
807:
710:(2003), "Spectra of random graphs with given expected degrees",
811:
309:
426:
299:
271:
243:
658:
420:
331:
218:
186:
166:
146:
107:
69:
1679:
1603:
1549:
1485:
1319:
1237:
1183:
1122:
846:
664:
627:
356:
314:
198:
172:
152:
129:
93:
380:(the number of incoming edges at each vertex) or
384:(the number of outgoing edges at each vertex).
823:
8:
1397:Fundamental (linear differential equation)
830:
816:
808:
749:
731:
657:
421:
419:
345:
330:
298:
270:
259:
242:
223:
217:
185:
165:
145:
116:
108:
106:
68:
394:
16:Type of matrix in algebraic graph theory
1702:Matrix representation of conic sections
692:
698:
696:
39:which contains information about the
7:
14:
1736:
779:Topics in algebraic graph theory
408:
1604:Used in science and engineering
847:Explicitly constrained entries
351:
338:
265:
252:
117:
109:
88:
76:
1:
1621:Fundamental (computer vision)
1387:Duplication and elimination
1186:eigenvalues or eigenvectors
652:has a constant diagonal of
357:{\displaystyle \deg(v_{i})}
1796:
1320:With specific applications
949:Discrete Fourier Transform
1730:
1611:Cabibbo–Kobayashi–Maskawa
1238:Satisfying conditions on
199:{\displaystyle n\times n}
969:Generalized permutation
733:10.1073/pnas.0937490100
648:The degree matrix of a
94:{\displaystyle G=(V,E)}
1775:Algebraic graph theory
1743:Mathematics portal
666:
629:
358:
316:
200:
174:
154:
131:
95:
25:algebraic graph theory
667:
630:
359:
317:
201:
175:
155:
132:
130:{\displaystyle |V|=n}
96:
656:
418:
399:Vertex labeled graph
376:may refer either to
329:
216:
184:
164:
144:
105:
67:
1692:Linear independence
939:Diagonally dominant
724:2003PNAS..100.6313C
1697:Matrix exponential
1687:Jordan normal form
1521:Fisher information
1392:Euclidean distance
1306:Totally unimodular
677:degree sum formula
662:
625:
619:
354:
312:
307:
303:
275:
196:
170:
150:
127:
91:
1762:
1761:
1754:Category:Matrices
1626:Fuzzy associative
1516:Doubly stochastic
1224:Positive-definite
904:Block tridiagonal
718:(11): 6313–6318,
675:According to the
665:{\displaystyle k}
638:
637:
325:where the degree
302:
279:
274:
173:{\displaystyle G}
153:{\displaystyle D}
51:to construct the
1787:
1749:List of matrices
1741:
1740:
1717:Row echelon form
1661:State transition
1590:Seidel adjacency
1472:Totally positive
1332:Alternating sign
929:Complex Hadamard
832:
825:
818:
809:
802:
800:
770:
764:
762:
753:
735:
700:
671:
669:
668:
663:
634:
632:
631:
626:
624:
623:
412:
395:
366:undirected graph
363:
361:
360:
355:
350:
349:
321:
319:
318:
313:
311:
308:
304:
300:
277:
276:
272:
264:
263:
234:
233:
205:
203:
202:
197:
179:
177:
176:
171:
159:
157:
156:
151:
136:
134:
133:
128:
120:
112:
100:
98:
97:
92:
53:Laplacian matrix
49:adjacency matrix
33:undirected graph
1795:
1794:
1790:
1789:
1788:
1786:
1785:
1784:
1765:
1764:
1763:
1758:
1735:
1726:
1675:
1599:
1545:
1481:
1315:
1233:
1179:
1118:
919:Centrosymmetric
842:
836:
806:
805:
790:
772:
771:
767:
706:; Lu, Linyuan;
702:
701:
694:
689:
654:
653:
650:k-regular graph
646:
618:
617:
612:
607:
602:
597:
592:
586:
585:
580:
575:
570:
565:
560:
554:
553:
548:
543:
538:
533:
528:
522:
521:
516:
511:
506:
501:
496:
490:
489:
484:
479:
474:
469:
464:
458:
457:
452:
447:
442:
437:
432:
422:
416:
415:
390:
341:
327:
326:
306:
305:
296:
290:
289:
268:
255:
238:
219:
214:
213:
207:diagonal matrix
182:
181:
162:
161:
142:
141:
103:
102:
65:
64:
61:
37:diagonal matrix
17:
12:
11:
5:
1793:
1791:
1783:
1782:
1777:
1767:
1766:
1760:
1759:
1757:
1756:
1751:
1746:
1731:
1728:
1727:
1725:
1724:
1719:
1714:
1709:
1707:Perfect matrix
1704:
1699:
1694:
1689:
1683:
1681:
1677:
1676:
1674:
1673:
1668:
1663:
1658:
1653:
1648:
1643:
1638:
1633:
1628:
1623:
1618:
1613:
1607:
1605:
1601:
1600:
1598:
1597:
1592:
1587:
1582:
1577:
1572:
1567:
1562:
1556:
1554:
1547:
1546:
1544:
1543:
1538:
1533:
1528:
1523:
1518:
1513:
1508:
1503:
1498:
1492:
1490:
1483:
1482:
1480:
1479:
1477:Transformation
1474:
1469:
1464:
1459:
1454:
1449:
1444:
1439:
1434:
1429:
1424:
1419:
1414:
1409:
1404:
1399:
1394:
1389:
1384:
1379:
1374:
1369:
1364:
1359:
1354:
1349:
1344:
1339:
1334:
1329:
1323:
1321:
1317:
1316:
1314:
1313:
1308:
1303:
1298:
1293:
1288:
1283:
1278:
1273:
1268:
1263:
1254:
1248:
1246:
1235:
1234:
1232:
1231:
1226:
1221:
1216:
1214:Diagonalizable
1211:
1206:
1201:
1196:
1190:
1188:
1184:Conditions on
1181:
1180:
1178:
1177:
1172:
1167:
1162:
1157:
1152:
1147:
1142:
1137:
1132:
1126:
1124:
1120:
1119:
1117:
1116:
1111:
1106:
1101:
1096:
1091:
1086:
1081:
1076:
1071:
1066:
1064:Skew-symmetric
1061:
1059:Skew-Hermitian
1056:
1051:
1046:
1041:
1036:
1031:
1026:
1021:
1016:
1011:
1006:
1001:
996:
991:
986:
981:
976:
971:
966:
961:
956:
951:
946:
941:
936:
931:
926:
921:
916:
911:
906:
901:
896:
894:Block-diagonal
891:
886:
881:
876:
871:
869:Anti-symmetric
866:
864:Anti-Hermitian
861:
856:
850:
848:
844:
843:
837:
835:
834:
827:
820:
812:
804:
803:
788:
765:
691:
690:
688:
685:
661:
645:
642:
636:
635:
622:
616:
613:
611:
608:
606:
603:
601:
598:
596:
593:
591:
588:
587:
584:
581:
579:
576:
574:
571:
569:
566:
564:
561:
559:
556:
555:
552:
549:
547:
544:
542:
539:
537:
534:
532:
529:
527:
524:
523:
520:
517:
515:
512:
510:
507:
505:
502:
500:
497:
495:
492:
491:
488:
485:
483:
480:
478:
475:
473:
470:
468:
465:
463:
460:
459:
456:
453:
451:
448:
446:
443:
441:
438:
436:
433:
431:
428:
427:
425:
413:
405:
404:
403:Degree matrix
401:
389:
386:
370:directed graph
353:
348:
344:
340:
337:
334:
323:
322:
310:
297:
295:
292:
291:
288:
285:
282:
269:
267:
262:
258:
254:
251:
248:
245:
244:
241:
237:
232:
229:
226:
222:
195:
192:
189:
169:
149:
126:
123:
119:
115:
111:
90:
87:
84:
81:
78:
75:
72:
63:Given a graph
60:
57:
15:
13:
10:
9:
6:
4:
3:
2:
1792:
1781:
1778:
1776:
1773:
1772:
1770:
1755:
1752:
1750:
1747:
1745:
1744:
1739:
1733:
1732:
1729:
1723:
1720:
1718:
1715:
1713:
1712:Pseudoinverse
1710:
1708:
1705:
1703:
1700:
1698:
1695:
1693:
1690:
1688:
1685:
1684:
1682:
1680:Related terms
1678:
1672:
1671:Z (chemistry)
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1632:
1629:
1627:
1624:
1622:
1619:
1617:
1614:
1612:
1609:
1608:
1606:
1602:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1566:
1563:
1561:
1558:
1557:
1555:
1553:
1548:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1497:
1494:
1493:
1491:
1489:
1484:
1478:
1475:
1473:
1470:
1468:
1465:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1408:
1405:
1403:
1400:
1398:
1395:
1393:
1390:
1388:
1385:
1383:
1380:
1378:
1375:
1373:
1370:
1368:
1365:
1363:
1360:
1358:
1355:
1353:
1350:
1348:
1345:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1324:
1322:
1318:
1312:
1309:
1307:
1304:
1302:
1299:
1297:
1294:
1292:
1289:
1287:
1284:
1282:
1279:
1277:
1274:
1272:
1269:
1267:
1264:
1262:
1258:
1255:
1253:
1250:
1249:
1247:
1245:
1241:
1236:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1191:
1189:
1187:
1182:
1176:
1173:
1171:
1168:
1166:
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1143:
1141:
1138:
1136:
1133:
1131:
1128:
1127:
1125:
1121:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1047:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1029:Pentadiagonal
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
985:
982:
980:
977:
975:
972:
970:
967:
965:
962:
960:
957:
955:
952:
950:
947:
945:
942:
940:
937:
935:
932:
930:
927:
925:
922:
920:
917:
915:
912:
910:
907:
905:
902:
900:
897:
895:
892:
890:
887:
885:
882:
880:
877:
875:
872:
870:
867:
865:
862:
860:
859:Anti-diagonal
857:
855:
852:
851:
849:
845:
840:
833:
828:
826:
821:
819:
814:
813:
810:
799:
795:
791:
789:0-521-80197-4
785:
781:
780:
775:
769:
766:
761:
757:
752:
747:
743:
739:
734:
729:
725:
721:
717:
713:
709:
705:
699:
697:
693:
686:
684:
682:
678:
673:
659:
651:
643:
641:
620:
614:
609:
604:
599:
594:
589:
582:
577:
572:
567:
562:
557:
550:
545:
540:
535:
530:
525:
518:
513:
508:
503:
498:
493:
486:
481:
476:
471:
466:
461:
454:
449:
444:
439:
434:
429:
423:
414:
411:
407:
406:
402:
400:
397:
396:
393:
387:
385:
383:
379:
375:
371:
367:
346:
342:
335:
332:
293:
286:
283:
280:
260:
256:
249:
246:
239:
235:
230:
227:
224:
220:
212:
211:
210:
208:
193:
190:
187:
167:
147:
140:
139:degree matrix
124:
121:
113:
85:
82:
79:
73:
70:
58:
56:
54:
50:
46:
42:
38:
34:
30:
29:degree matrix
26:
22:
1734:
1666:Substitution
1569:
1552:graph theory
1049:Quaternionic
1039:Persymmetric
778:
774:Mohar, Bojan
768:
715:
711:
674:
647:
639:
391:
373:
324:
138:
62:
28:
21:mathematical
18:
1641:Hamiltonian
1565:Biadjacency
1501:Correlation
1417:Householder
1367:Commutation
1104:Vandermonde
1099:Tridiagonal
1034:Permutation
1024:Nonnegative
1009:Matrix unit
889:Bisymmetric
372:, the term
209:defined as
1769:Categories
1541:Transition
1536:Stochastic
1506:Covariance
1488:statistics
1467:Symplectic
1462:Similarity
1291:Unimodular
1286:Orthogonal
1271:Involutory
1266:Invertible
1261:Projection
1257:Idempotent
1199:Convergent
1094:Triangular
1044:Polynomial
989:Hessenberg
959:Equivalent
954:Elementary
934:Copositive
924:Conference
884:Bidiagonal
704:Chung, Fan
687:References
644:Properties
59:Definition
1722:Wronskian
1646:Irregular
1636:Gell-Mann
1585:Laplacian
1580:Incidence
1560:Adjacency
1531:Precision
1496:Centering
1402:Generator
1372:Confusion
1357:Circulant
1337:Augmented
1296:Unipotent
1276:Nilpotent
1252:Congruent
1229:Stieltjes
1204:Defective
1194:Companion
1165:Redheffer
1084:Symmetric
1079:Sylvester
1054:Signature
984:Hermitian
964:Frobenius
874:Arrowhead
854:Alternant
382:outdegree
336:
301:otherwise
250:
191:×
23:field of
1780:Matrices
1550:Used in
1486:Used in
1447:Rotation
1422:Jacobian
1382:Distance
1362:Cofactor
1347:Carleman
1327:Adjugate
1311:Weighing
1244:inverses
1240:products
1209:Definite
1140:Identity
1130:Exchange
1123:Constant
1089:Toeplitz
974:Hadamard
944:Diagonal
760:12743375
378:indegree
43:of each
1651:Overlap
1616:Density
1575:Edmonds
1452:Seifert
1412:Hessian
1377:Coxeter
1301:Unitary
1219:Hurwitz
1150:Of ones
1135:Hilbert
1069:Skyline
1014:Metzler
1004:Logical
999:Integer
909:Boolean
841:classes
798:2125091
742:1982145
720:Bibcode
708:Vu, Van
388:Example
19:In the
1570:Degree
1511:Design
1442:Random
1432:Payoff
1427:Moment
1352:Cartan
1342:BĂ©zout
1281:Normal
1155:Pascal
1145:Lehmer
1074:Sparse
994:Hollow
979:Hankel
914:Cauchy
839:Matrix
796:
786:
758:
751:164443
748:
740:
679:, the
374:degree
278:
137:, the
45:vertex
41:degree
31:of an
27:, the
1631:Gamma
1595:Tutte
1457:Shear
1170:Shift
1160:Pauli
1109:Walsh
1019:Moore
899:Block
681:trace
180:is a
101:with
35:is a
1437:Pick
1407:Gram
1175:Zero
879:Band
784:ISBN
756:PMID
160:for
1526:Hat
1259:or
1242:or
746:PMC
728:doi
716:100
333:deg
247:deg
1771::
794:MR
792:,
754:,
744:,
738:MR
736:,
726:,
714:,
695:^
672:.
273:if
236::=
1656:S
1114:Z
831:e
824:t
817:v
801:.
763:.
730::
722::
660:k
621:)
615:1
610:0
605:0
600:0
595:0
590:0
583:0
578:3
573:0
568:0
563:0
558:0
551:0
546:0
541:3
536:0
531:0
526:0
519:0
514:0
509:0
504:2
499:0
494:0
487:0
482:0
477:0
472:0
467:3
462:0
455:0
450:0
445:0
440:0
435:0
430:4
424:(
352:)
347:i
343:v
339:(
294:0
287:j
284:=
281:i
266:)
261:i
257:v
253:(
240:{
231:j
228:,
225:i
221:D
194:n
188:n
168:G
148:D
125:n
122:=
118:|
114:V
110:|
89:)
86:E
83:,
80:V
77:(
74:=
71:G
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
