604:
542:
204:
898:
660:
245:
390:
783:
746:
710:
458:
355:
319:
278:
143:
1055:
107:
968:
1008:
988:
941:
921:
834:
814:
425:
554:
480:
156:
400:
of the vertex space is thus the number of vertices of the graph, while the dimension of the edge space is the number of edges.
845:
610:
209:
1097:
1089:
1133:
360:
17:
755:
718:
682:
430:
327:
291:
250:
115:
403:
These definitions can be made more explicit. For example, we can describe the edge space as follows:
397:
944:
668:
1016:
752:
made into a vector space with similar vector addition and scalar multiplication as defined for
676:
74:
837:
546:
56:
474:
8:
52:
950:
993:
973:
926:
906:
819:
799:
410:
1093:
794:
44:
1069:
470:
60:
59:
sets, respectively. These vector spaces make it possible to use techniques of
1127:
150:
48:
32:
28:
1113:
1061:
1118:
1065:
461:
1083:
1103:(the electronic 3rd edition is freely available on author's site).
599:{\displaystyle 0\cdot P:=\emptyset \qquad P\in {\mathcal {E}}(G)}
537:{\displaystyle P+Q:=P\triangle Q\qquad P,Q\in {\mathcal {E}}(G)}
199:{\displaystyle \mathbb {Z} /2\mathbb {Z} :=\lbrace 0,1\rbrace }
284:
which assigns a 1 to its vertices. Also every subset of
943:, as a linear transformation, maps each edge to its two
893:{\displaystyle H:{\mathcal {E}}(G)\to {\mathcal {V}}(G)}
655:{\displaystyle 1\cdot P:=P\qquad P\in {\mathcal {E}}(G)}
240:{\displaystyle V\rightarrow \mathbb {Z} /2\mathbb {Z} }
1019:
996:
976:
953:
929:
909:
848:
822:
802:
758:
721:
685:
613:
557:
483:
433:
413:
363:
330:
294:
253:
212:
159:
118:
77:
1049:
1002:
982:
962:
935:
915:
892:
828:
808:
777:
740:
704:
654:
598:
536:
452:
419:
384:
349:
313:
272:
239:
198:
137:
101:
1125:
903:between the edge space and the vertex space of
392:-vector space freely generated by the edge set
193:
181:
385:{\displaystyle \mathbb {Z} /2\mathbb {Z} }
378:
365:
233:
220:
174:
161:
1081:
14:
1126:
321:by its characteristic function. The
280:naturally corresponds the subset of
24:
876:
857:
761:
724:
688:
638:
582:
570:
520:
499:
436:
333:
297:
256:
121:
109:be a finite undirected graph. The
25:
1145:
778:{\displaystyle {\mathcal {E}}(G)}
741:{\displaystyle {\mathcal {V}}(G)}
705:{\displaystyle {\mathcal {E}}(G)}
453:{\displaystyle {\mathcal {E}}(G)}
350:{\displaystyle {\mathcal {E}}(G)}
314:{\displaystyle {\mathcal {V}}(G)}
273:{\displaystyle {\mathcal {V}}(G)}
138:{\displaystyle {\mathcal {V}}(G)}
629:
573:
505:
1032:
1023:
887:
881:
871:
868:
862:
772:
766:
735:
729:
699:
693:
649:
643:
593:
587:
531:
525:
447:
441:
344:
338:
308:
302:
267:
261:
216:
132:
126:
96:
84:
13:
1:
1075:
788:
149:is the vector space over the
66:
7:
1107:
288:is uniquely represented in
10:
1150:
1082:Diestel, Reinhard (2005),
923:. The incidence matrix of
1050:{\displaystyle H(vu)=v+u}
407:elements are subsets of
102:{\displaystyle G:=(V,E)}
51:defined in terms of the
63:in studying the graph.
1134:Algebraic graph theory
1051:
1004:
984:
964:
937:
917:
894:
830:
810:
779:
742:
715:One can also think of
706:
656:
600:
538:
454:
421:
386:
351:
315:
274:
241:
200:
139:
103:
1052:
1005:
985:
965:
938:
918:
895:
838:linear transformation
836:defines one possible
831:
811:
780:
743:
707:
657:
601:
547:scalar multiplication
539:
455:
422:
387:
352:
316:
275:
242:
201:
140:
104:
1017:
994:
974:
970:be the edge between
951:
927:
907:
846:
820:
800:
756:
748:as the power set of
719:
683:
611:
555:
481:
475:symmetric difference
431:
427:, that is, as a set
411:
361:
328:
292:
251:
247:. Every element of
210:
157:
116:
75:
1072:of the edge space.
1047:
1000:
980:
963:{\displaystyle vu}
960:
933:
913:
890:
826:
806:
775:
738:
702:
652:
596:
534:
473:is defined as the
450:
417:
382:
347:
311:
270:
237:
196:
135:
99:
1003:{\displaystyle u}
983:{\displaystyle v}
936:{\displaystyle G}
916:{\displaystyle G}
829:{\displaystyle G}
809:{\displaystyle H}
420:{\displaystyle E}
206:of all functions
153:of two elements
16:(Redirected from
1141:
1102:
1088:(3rd ed.),
1056:
1054:
1053:
1048:
1009:
1007:
1006:
1001:
989:
987:
986:
981:
969:
967:
966:
961:
942:
940:
939:
934:
922:
920:
919:
914:
899:
897:
896:
891:
880:
879:
861:
860:
835:
833:
832:
827:
815:
813:
812:
807:
795:incidence matrix
784:
782:
781:
776:
765:
764:
747:
745:
744:
739:
728:
727:
711:
709:
708:
703:
692:
691:
661:
659:
658:
653:
642:
641:
605:
603:
602:
597:
586:
585:
543:
541:
540:
535:
524:
523:
459:
457:
456:
451:
440:
439:
426:
424:
423:
418:
391:
389:
388:
383:
381:
373:
368:
356:
354:
353:
348:
337:
336:
320:
318:
317:
312:
301:
300:
279:
277:
276:
271:
260:
259:
246:
244:
243:
238:
236:
228:
223:
205:
203:
202:
197:
177:
169:
164:
144:
142:
141:
136:
125:
124:
108:
106:
105:
100:
45:undirected graph
21:
1149:
1148:
1144:
1143:
1142:
1140:
1139:
1138:
1124:
1123:
1110:
1100:
1078:
1018:
1015:
1014:
995:
992:
991:
975:
972:
971:
952:
949:
948:
928:
925:
924:
908:
905:
904:
875:
874:
856:
855:
847:
844:
843:
821:
818:
817:
801:
798:
797:
791:
760:
759:
757:
754:
753:
723:
722:
720:
717:
716:
687:
686:
684:
681:
680:
637:
636:
612:
609:
608:
581:
580:
556:
553:
552:
549:is defined by:
519:
518:
482:
479:
478:
471:vector addition
435:
434:
432:
429:
428:
412:
409:
408:
377:
369:
364:
362:
359:
358:
332:
331:
329:
326:
325:
296:
295:
293:
290:
289:
255:
254:
252:
249:
248:
232:
224:
219:
211:
208:
207:
173:
165:
160:
158:
155:
154:
120:
119:
117:
114:
113:
76:
73:
72:
69:
23:
22:
15:
12:
11:
5:
1147:
1137:
1136:
1122:
1121:
1116:
1109:
1106:
1105:
1104:
1098:
1077:
1074:
1058:
1057:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
999:
979:
959:
956:
947:vertices. Let
932:
912:
901:
900:
889:
886:
883:
878:
873:
870:
867:
864:
859:
854:
851:
825:
805:
790:
787:
774:
771:
768:
763:
737:
734:
731:
726:
701:
698:
695:
690:
665:
664:
663:
662:
651:
648:
645:
640:
635:
632:
628:
625:
622:
619:
616:
606:
595:
592:
589:
584:
579:
576:
572:
569:
566:
563:
560:
544:
533:
530:
527:
522:
517:
514:
511:
508:
504:
501:
498:
495:
492:
489:
486:
468:
449:
446:
443:
438:
416:
380:
376:
372:
367:
346:
343:
340:
335:
310:
307:
304:
299:
269:
266:
263:
258:
235:
231:
227:
222:
218:
215:
195:
192:
189:
186:
183:
180:
176:
172:
168:
163:
134:
131:
128:
123:
98:
95:
92:
89:
86:
83:
80:
68:
65:
61:linear algebra
31:discipline of
9:
6:
4:
3:
2:
1146:
1135:
1132:
1131:
1129:
1120:
1117:
1115:
1112:
1111:
1101:
1099:3-540-26182-6
1095:
1091:
1087:
1086:
1080:
1079:
1073:
1071:
1067:
1063:
1044:
1041:
1038:
1035:
1029:
1026:
1020:
1013:
1012:
1011:
997:
977:
957:
954:
946:
930:
910:
884:
865:
852:
849:
842:
841:
840:
839:
823:
803:
796:
786:
769:
751:
732:
713:
696:
678:
674:
670:
646:
633:
630:
626:
623:
620:
617:
614:
607:
590:
577:
574:
567:
564:
561:
558:
551:
550:
548:
545:
528:
515:
512:
509:
506:
502:
496:
493:
490:
487:
484:
476:
472:
469:
467:
463:
444:
414:
406:
405:
404:
401:
399:
395:
374:
370:
341:
324:
305:
287:
283:
264:
229:
225:
213:
190:
187:
184:
178:
170:
166:
152:
148:
129:
112:
93:
90:
87:
81:
78:
64:
62:
58:
54:
50:
49:vector spaces
46:
42:
38:
34:
30:
19:
1085:Graph Theory
1084:
1059:
902:
816:for a graph
792:
749:
714:
672:
666:
465:
402:
393:
322:
285:
281:
151:finite field
146:
111:vertex space
110:
70:
41:vertex space
40:
36:
33:graph theory
29:mathematical
26:
18:Vertex space
1114:Cycle space
1062:cycle space
671:subsets of
37:edge space
1076:References
789:Properties
323:edge space
67:Definition
1119:Cut space
1070:subspaces
1066:cut space
872:→
669:singleton
634:∈
618:⋅
578:∈
571:∅
562:⋅
516:∈
500:△
462:power set
398:dimension
217:→
1128:Category
1108:See also
1090:Springer
1064:and the
945:incident
675:form a
460:is the
357:is the
27:In the
1096:
396:. The
57:vertex
43:of an
35:, the
1010:then
677:basis
1094:ISBN
1068:are
1060:The
990:and
793:The
679:for
667:The
71:Let
55:and
53:edge
47:are
39:and
464:of
145:of
1130::
1092:,
785:.
712:.
624::=
568::=
494::=
477::
179::=
82::=
1045:u
1042:+
1039:v
1036:=
1033:)
1030:u
1027:v
1024:(
1021:H
998:u
978:v
958:u
955:v
931:G
911:G
888:)
885:G
882:(
877:V
869:)
866:G
863:(
858:E
853::
850:H
824:G
804:H
773:)
770:G
767:(
762:E
750:V
736:)
733:G
730:(
725:V
700:)
697:G
694:(
689:E
673:E
650:)
647:G
644:(
639:E
631:P
627:P
621:P
615:1
594:)
591:G
588:(
583:E
575:P
565:P
559:0
532:)
529:G
526:(
521:E
513:Q
510:,
507:P
503:Q
497:P
491:Q
488:+
485:P
466:E
448:)
445:G
442:(
437:E
415:E
394:E
379:Z
375:2
371:/
366:Z
345:)
342:G
339:(
334:E
309:)
306:G
303:(
298:V
286:V
282:V
268:)
265:G
262:(
257:V
234:Z
230:2
226:/
221:Z
214:V
194:}
191:1
188:,
185:0
182:{
175:Z
171:2
167:/
162:Z
147:G
133:)
130:G
127:(
122:V
97:)
94:E
91:,
88:V
85:(
79:G
20:)
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