3077:
1136:
107:
802:
642:
279:
1110:
that orients the given signed graph. The column of a positive edge has a 1 in the row corresponding to one endpoint and a â1 in the row corresponding to the other endpoint, just like an edge in an ordinary (unsigned) graph. The column of a negative edge has either a 1 or a â1 in both rows. The line
1372:
579:
797:{\displaystyle B_{ij}=\left\{{\begin{array}{rl}{-1}&{\text{if edge }}e_{j}{\text{ leaves vertex }}v_{i},\\{\phantom {-}}1&{\text{if edge }}e_{j}{\text{ enters vertex }}v_{i},\\{\phantom {-}}0&{\text{otherwise.}}\end{array}}\right.}
183:
1127:. The column of an oriented incidence matrix that corresponds to a loop is all zero, unless the graph is signed and the loop is negative; then the column is all zero except for ±2 in the row of its incident vertex.
1257:
464:
284:
For example, the incidence matrix of the undirected graph shown on the right is a matrix consisting of 4 rows (corresponding to the four vertices, 1â4) and 4 columns (corresponding to the four edges,
950:
1562:, called "blocks", subject to rules that depend on the type of design. The incidence matrix is an important tool in the theory of block designs. For instance, it can be used to prove
1065:
348:
1385:
Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries. By contrast, a
2099:
1566:, a fundamental theorem of balanced incomplete 2-designs (BIBDs), that the number of blocks is at least the number of points. Considering the blocks as a system of sets, the
2735:
622:
153:
1731:
274:{\displaystyle B_{ij}=\left\{{\begin{array}{rl}\,1&{\text{if vertex }}v_{i}{\text{ is incident with edge }}e_{j},\\0&{\text{otherwise.}}\end{array}}\right.}
1907:
2949:
2168:
587:
If we look at the incidence matrix, we see that the sum of each column is equal to 2. This is because each edge has a vertex connected to each end.
3040:
1143:
A weighted graph can be represented using the weight of the edge in place of a 1. For example, the incidence matrix of the graph to the right is:
1538:
In a similar manner, the relationship between cells whose dimensions differ by one in a polytope, can be represented by an incidence matrix.
2959:
2725:
2114:
2068:
1724:
2058:
2028:
1900:
1571:
1668:
1621:
1367:{\displaystyle {\begin{bmatrix}2&1&5&0\\2&0&0&0\\0&1&0&6\\0&0&5&6\\\end{bmatrix}}.}
1090:. The binary cycle space is the null space of its oriented or unoriented incidence matrix, viewed as a matrix over the two-element
574:{\displaystyle {\begin{bmatrix}1&1&1&0\\1&0&0&0\\0&1&0&1\\0&0&1&1\\\end{bmatrix}}.}
2104:
1825:
3113:
2760:
2135:
1514:
could be the set of subspaces of dimension one less than the dimension of the entire space (hyperplanes); or, more generally,
3128:
2307:
1717:
862:
834:
negation of any of the columns, since negating the entries of a column corresponds to reversing the orientation of an edge.
2078:
1893:
1660:
2094:
1583:
1389:
can have multiple vertices assigned to one edge; thus, a general matrix of non-negative integers describes a hypergraph.
2524:
2161:
2109:
2599:
2755:
2277:
1752:
815:
2859:
2730:
2644:
2964:
2854:
2562:
2242:
2010:
1830:
1608:
1680:, second edition, 2006 (p 97, Incidence Matrices for undirected graphs; p 98, incidence matrices for digraphs)
1017:
287:
2999:
2928:
2810:
2670:
2267:
2154:
2043:
1567:
3118:
2869:
2452:
2257:
2053:
1563:
2815:
2552:
2402:
2397:
2232:
2207:
2202:
2130:
1998:
170:
3076:
3009:
2367:
2197:
2177:
2063:
2033:
1973:
1951:
1848:
155:
43:
3030:
3004:
2582:
2387:
2377:
2073:
2038:
2018:
1916:
1402:
1120:
1091:
174:
3081:
3035:
3025:
2979:
2974:
2903:
2839:
2705:
2442:
2437:
2372:
2362:
2227:
2048:
1968:
1471:
1613:
601:
132:
3123:
3092:
2879:
2874:
2864:
2844:
2805:
2800:
2629:
2624:
2609:
2604:
2595:
2590:
2537:
2432:
2382:
2327:
2297:
2292:
2272:
2262:
2222:
1692:
1664:
1617:
3087:
3055:
2984:
2923:
2898:
2834:
2740:
2710:
2695:
2680:
2675:
2614:
2567:
2542:
2532:
2503:
2422:
2417:
2392:
2322:
2302:
2212:
2192:
1978:
1767:
1640:, The Carus Mathematical Monographs #14, The Mathematical Association of America, p. 99
1107:
1000:
842:
115:
95:
1135:
2785:
2720:
2700:
2685:
2665:
2649:
2547:
2478:
2468:
2427:
2312:
2282:
1495:
1087:
989:
106:
1106:
is a generalization of the oriented incidence matrix. It is the incidence matrix of any
814:
of an undirected graph is the incidence matrix, in the sense of directed graphs, of any
3045:
2989:
2969:
2954:
2913:
2790:
2750:
2715:
2639:
2578:
2557:
2498:
2488:
2473:
2407:
2352:
2342:
2337:
2247:
1961:
1864:
1757:
1124:
595:
39:
3107:
3050:
2908:
2849:
2780:
2770:
2765:
2690:
2619:
2493:
2483:
2412:
2332:
2317:
2252:
1869:
1842:
1695:
2933:
2890:
2795:
2508:
2447:
2357:
2237:
1956:
1836:
1547:
1103:
1083:
91:
17:
2775:
2745:
2513:
2347:
2217:
1874:
1071:
31:
667:
208:
2826:
2287:
2023:
1988:
1935:
1479:
1475:
1386:
1075:
846:
42:
that shows the relationship between two classes of objects, usually called an
3060:
2634:
1762:
1700:
1470:
are incident and 0 otherwise. In this case, the incidence matrix is also a
2994:
1486:, the incidence matrix of an incidence structure describes a hypergraph.
1079:
82:
in this context) and 0 if they are not. There are variations; see below.
1709:
1885:
1799:
1603:
1003:(or Kirchhoff matrix) is obtained from the oriented incidence matrix
830:, and all other rows have 0. The oriented incidence matrix is unique
1111:
graph and
Kirchhoff matrix properties generalize to signed graphs.
1809:
1134:
831:
2146:
27:
Matrix that shows the relationship between two classes of objects
1788:
2150:
1889:
1713:
1506:
is the set of lines. In a finite geometry of higher dimension,
1804:
1078:
of its oriented incidence matrix, viewed as a matrix over the
118:
has two kinds of incidence matrices: unoriented and oriented.
636:
are the number of vertices and edges respectively, such that
826:
and one â1 in the row corresponding to the other vertex of
822:, there is one 1 in the row corresponding to one vertex of
791:
268:
1119:
The definitions of incidence matrix apply to graphs with
1266:
945:{\displaystyle A(L(G))=B(G)^{\textsf {T}}B(G)-2I_{m}.}
473:
1260:
1149:
1020:
865:
645:
604:
467:
356:
290:
186:
135:
98:, which encodes the relation of vertex-vertex pairs.
90:
Incidence matrix is a common graph representation in
3018:
2942:
2888:
2824:
2658:
2576:
2522:
2461:
2185:
2123:
2087:
2009:
1944:
1923:
1857:
1818:
1781:
1745:
1518:could be the set of all subspaces of one dimension
1366:
1059:
944:
796:
616:
573:
342:
273:
147:
1663:, vol. 173 (3rd ed.), Springer-Verlag,
807:(Many authors use the opposite sign convention.)
967:)) is the adjacency matrix of the line graph of
1526:the set of all subspaces of another dimension
2162:
1901:
1725:
818:of the graph. That is, in the column of edge
54:, the matrix has one row for each element of
8:
837:The unoriented incidence matrix of a graph
2736:Fundamental (linear differential equation)
2169:
2155:
2147:
1908:
1894:
1886:
1732:
1718:
1710:
1570:of the incidence matrix is the number of
1530:, with incidence defined as containment.
1261:
1259:
1048:
1047:
1046:
1019:
933:
905:
904:
903:
864:
782:
769:
768:
755:
746:
740:
731:
718:
717:
704:
695:
689:
680:
670:
666:
650:
644:
603:
468:
466:
334:
321:
308:
295:
289:
259:
241:
232:
226:
217:
211:
207:
191:
185:
134:
105:
3041:Matrix representation of conic sections
1595:
1060:{\displaystyle B(G)B(G)^{\textsf {T}}.}
343:{\displaystyle e_{1},e_{2},e_{3},e_{4}}
7:
1572:systems of distinct representatives
1498:. For instance, in a finite plane,
58:and one column for each element of
25:
1678:Graph Theory and its applications
234: is incident with edge
3075:
1612:(3rd ed.), Dover, pp.
1554:is a finite set of "points" and
1478:of the structure. As there is a
2943:Used in science and engineering
1510:could be the set of points and
979:) is the incidence matrix, and
2186:Explicitly constrained entries
2059:CremonaâRichmond configuration
1676:Jonathan L Gross, Jay Yellen,
1043:
1036:
1030:
1024:
920:
914:
900:
893:
884:
881:
875:
869:
129:) of an undirected graph is a
102:Undirected and directed graphs
1:
2960:Fundamental (computer vision)
1661:Graduate Texts in Mathematics
2136:Kirkman's schoolgirl problem
2069:GrĂŒnbaumâRigby configuration
1839:for cubic Hamiltonian graphs
1636:Ryser, Herbert John (1963),
1098:Signed and bidirected graphs
856:) by the following theorem:
2726:Duplication and elimination
2525:eigenvalues or eigenvectors
2029:MöbiusâKantor configuration
1139:A weighted undirected graph
1074:of a graph is equal to the
123:unoriented incidence matrix
3145:
2659:With specific applications
2288:Discrete Fourier Transform
2115:BruckâRyserâChowla theorem
1753:Graph (abstract data type)
1655:Diestel, Reinhard (2005),
1494:An important example is a
1482:for every Levi graph, and
1422:(or its transpose), where
1102:The incidence matrix of a
3069:
2950:CabibboâKobayashiâMaskawa
2577:Satisfying conditions on
2105:SzemerĂ©diâTrotter theorem
1638:Combinatorial Mathematics
1558:is a class of subsets of
1502:is the set of points and
812:oriented incidence matrix
748: enters vertex
697: leaves vertex
617:{\displaystyle n\times m}
148:{\displaystyle n\times m}
2095:SylvesterâGallai theorem
1831:Graph Modelling Language
1584:ParryâSullivan invariant
1438:respectively, such that
177:respectively, such that
94:. It is different to an
46:. If the first class is
2308:Generalized permutation
2100:De BruijnâErdĆs theorem
2044:Desargues configuration
3114:Algebraic graph theory
3082:Mathematics portal
1368:
1140:
1061:
946:
798:
618:
575:
344:
275:
149:
111:
3129:Graph data structures
2131:Design of experiments
1740:Graph representations
1546:Another example is a
1369:
1138:
1062:
947:
799:
619:
576:
345:
276:
150:
109:
2064:Kummer configuration
2034:Pappus configuration
1917:Incidence structures
1849:Trivial Graph Format
1393:Incidence structures
1258:
1018:
863:
774:
723:
643:
602:
465:
288:
184:
133:
110:An undirected graph.
78:are related (called
3031:Linear independence
2278:Diagonally dominant
2074:Klein configuration
2054:SchlÀfli double six
2039:Hesse configuration
2019:Complete quadrangle
1564:Fisher's inequality
1403:incidence structure
770:
719:
169:are the numbers of
114:In graph theory an
62:. The entry in row
3036:Matrix exponential
3026:Jordan normal form
2860:Fisher information
2731:Euclidean distance
2645:Totally unimodular
2049:Reye configuration
1819:Text-based formats
1696:"Incidence matrix"
1693:Weisstein, Eric W.
1472:biadjacency matrix
1430:are the number of
1364:
1355:
1141:
1057:
942:
841:is related to the
794:
789:
614:
571:
562:
340:
271:
266:
145:
112:
50:and the second is
44:incidence relation
18:Incidence relation
3101:
3100:
3093:Category:Matrices
2965:Fuzzy associative
2855:Doubly stochastic
2563:Positive-definite
2243:Block tridiagonal
2144:
2143:
1883:
1882:
1782:XML-based formats
1609:Regular Polytopes
1490:Finite geometries
1378:
1377:
1248:
1247:
1050:
1011:) by the formula
907:
785:
749:
734:
698:
683:
585:
584:
455:
454:
262:
235:
220:
16:(Redirected from
3136:
3088:List of matrices
3080:
3079:
3056:Row echelon form
3000:State transition
2929:Seidel adjacency
2811:Totally positive
2671:Alternating sign
2268:Complex Hadamard
2171:
2164:
2157:
2148:
1979:Projective plane
1931:Incidence matrix
1910:
1903:
1896:
1887:
1858:Related concepts
1773:Incidence matrix
1768:Adjacency matrix
1734:
1727:
1720:
1711:
1706:
1705:
1673:
1642:
1641:
1633:
1627:
1626:
1600:
1453:
1417:
1399:incidence matrix
1373:
1371:
1370:
1365:
1360:
1359:
1150:
1146:
1145:
1108:bidirected graph
1066:
1064:
1063:
1058:
1053:
1052:
1051:
951:
949:
948:
943:
938:
937:
910:
909:
908:
843:adjacency matrix
803:
801:
800:
795:
793:
790:
786:
783:
776:
775:
760:
759:
750:
747:
745:
744:
735:
732:
725:
724:
709:
708:
699:
696:
694:
693:
684:
681:
677:
658:
657:
623:
621:
620:
615:
592:incidence matrix
580:
578:
577:
572:
567:
566:
357:
353:
352:
349:
347:
346:
341:
339:
338:
326:
325:
313:
312:
300:
299:
280:
278:
277:
272:
270:
267:
263:
260:
246:
245:
236:
233:
231:
230:
221:
218:
199:
198:
154:
152:
151:
146:
127:incidence matrix
116:undirected graph
96:adjacency matrix
36:incidence matrix
21:
3144:
3143:
3139:
3138:
3137:
3135:
3134:
3133:
3104:
3103:
3102:
3097:
3074:
3065:
3014:
2938:
2884:
2820:
2654:
2572:
2518:
2457:
2258:Centrosymmetric
2181:
2175:
2145:
2140:
2119:
2083:
2005:
1940:
1936:Incidence graph
1919:
1914:
1884:
1879:
1853:
1814:
1777:
1746:Data structures
1741:
1738:
1691:
1690:
1687:
1671:
1654:
1651:
1649:Further reading
1646:
1645:
1635:
1634:
1630:
1624:
1604:Coxeter, H.S.M.
1602:
1601:
1597:
1592:
1580:
1544:
1536:
1496:finite geometry
1492:
1469:
1460:
1451:
1439:
1409:
1395:
1383:
1354:
1353:
1348:
1343:
1338:
1332:
1331:
1326:
1321:
1316:
1310:
1309:
1304:
1299:
1294:
1288:
1287:
1282:
1277:
1272:
1262:
1256:
1255:
1176:
1170:
1164:
1158:
1133:
1131:Weighted graphs
1117:
1100:
1088:complex numbers
1042:
1016:
1015:
990:identity matrix
987:
929:
899:
861:
860:
788:
787:
780:
765:
764:
751:
736:
729:
714:
713:
700:
685:
678:
662:
646:
641:
640:
600:
599:
561:
560:
555:
550:
545:
539:
538:
533:
528:
523:
517:
516:
511:
506:
501:
495:
494:
489:
484:
479:
469:
463:
462:
383:
377:
371:
365:
330:
317:
304:
291:
286:
285:
265:
264:
257:
251:
250:
237:
222:
219:if vertex
215:
203:
187:
182:
181:
131:
130:
104:
88:
28:
23:
22:
15:
12:
11:
5:
3142:
3140:
3132:
3131:
3126:
3121:
3116:
3106:
3105:
3099:
3098:
3096:
3095:
3090:
3085:
3070:
3067:
3066:
3064:
3063:
3058:
3053:
3048:
3046:Perfect matrix
3043:
3038:
3033:
3028:
3022:
3020:
3016:
3015:
3013:
3012:
3007:
3002:
2997:
2992:
2987:
2982:
2977:
2972:
2967:
2962:
2957:
2952:
2946:
2944:
2940:
2939:
2937:
2936:
2931:
2926:
2921:
2916:
2911:
2906:
2901:
2895:
2893:
2886:
2885:
2883:
2882:
2877:
2872:
2867:
2862:
2857:
2852:
2847:
2842:
2837:
2831:
2829:
2822:
2821:
2819:
2818:
2816:Transformation
2813:
2808:
2803:
2798:
2793:
2788:
2783:
2778:
2773:
2768:
2763:
2758:
2753:
2748:
2743:
2738:
2733:
2728:
2723:
2718:
2713:
2708:
2703:
2698:
2693:
2688:
2683:
2678:
2673:
2668:
2662:
2660:
2656:
2655:
2653:
2652:
2647:
2642:
2637:
2632:
2627:
2622:
2617:
2612:
2607:
2602:
2593:
2587:
2585:
2574:
2573:
2571:
2570:
2565:
2560:
2555:
2553:Diagonalizable
2550:
2545:
2540:
2535:
2529:
2527:
2523:Conditions on
2520:
2519:
2517:
2516:
2511:
2506:
2501:
2496:
2491:
2486:
2481:
2476:
2471:
2465:
2463:
2459:
2458:
2456:
2455:
2450:
2445:
2440:
2435:
2430:
2425:
2420:
2415:
2410:
2405:
2403:Skew-symmetric
2400:
2398:Skew-Hermitian
2395:
2390:
2385:
2380:
2375:
2370:
2365:
2360:
2355:
2350:
2345:
2340:
2335:
2330:
2325:
2320:
2315:
2310:
2305:
2300:
2295:
2290:
2285:
2280:
2275:
2270:
2265:
2260:
2255:
2250:
2245:
2240:
2235:
2233:Block-diagonal
2230:
2225:
2220:
2215:
2210:
2208:Anti-symmetric
2205:
2203:Anti-Hermitian
2200:
2195:
2189:
2187:
2183:
2182:
2176:
2174:
2173:
2166:
2159:
2151:
2142:
2141:
2139:
2138:
2133:
2127:
2125:
2121:
2120:
2118:
2117:
2112:
2110:Beck's theorem
2107:
2102:
2097:
2091:
2089:
2085:
2084:
2082:
2081:
2076:
2071:
2066:
2061:
2056:
2051:
2046:
2041:
2036:
2031:
2026:
2021:
2015:
2013:
2011:Configurations
2007:
2006:
2004:
2003:
2002:
2001:
1993:
1992:
1991:
1983:
1982:
1981:
1976:
1966:
1965:
1964:
1962:Steiner system
1959:
1948:
1946:
1942:
1941:
1939:
1938:
1933:
1927:
1925:
1924:Representation
1921:
1920:
1915:
1913:
1912:
1905:
1898:
1890:
1881:
1880:
1878:
1877:
1872:
1867:
1865:Graph database
1861:
1859:
1855:
1854:
1852:
1851:
1846:
1840:
1834:
1828:
1822:
1820:
1816:
1815:
1813:
1812:
1807:
1802:
1797:
1794:
1791:
1785:
1783:
1779:
1778:
1776:
1775:
1770:
1765:
1760:
1758:Adjacency list
1755:
1749:
1747:
1743:
1742:
1739:
1737:
1736:
1729:
1722:
1714:
1708:
1707:
1686:
1685:External links
1683:
1682:
1681:
1674:
1669:
1650:
1647:
1644:
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1628:
1622:
1594:
1593:
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1579:
1576:
1543:
1540:
1535:
1532:
1491:
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1465:
1458:
1443:
1394:
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1265:
1252:
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1239:
1236:
1233:
1229:
1228:
1225:
1222:
1219:
1216:
1212:
1211:
1208:
1205:
1202:
1199:
1195:
1194:
1191:
1188:
1185:
1182:
1178:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1132:
1129:
1125:multiple edges
1116:
1113:
1099:
1096:
1068:
1067:
1056:
1045:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
983:
953:
952:
941:
936:
932:
928:
925:
922:
919:
916:
913:
902:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
805:
804:
792:
781:
779:
773:
767:
766:
763:
758:
754:
743:
739:
730:
728:
722:
716:
715:
712:
707:
703:
692:
688:
679:
676:
673:
669:
668:
665:
661:
656:
653:
649:
613:
610:
607:
596:directed graph
583:
582:
570:
565:
559:
556:
554:
551:
549:
546:
544:
541:
540:
537:
534:
532:
529:
527:
524:
522:
519:
518:
515:
512:
510:
507:
505:
502:
500:
497:
496:
493:
490:
488:
485:
483:
480:
478:
475:
474:
472:
459:
456:
453:
452:
449:
446:
443:
440:
436:
435:
432:
429:
426:
423:
419:
418:
415:
412:
409:
406:
402:
401:
398:
395:
392:
389:
385:
384:
381:
378:
375:
372:
369:
366:
363:
360:
337:
333:
329:
324:
320:
316:
311:
307:
303:
298:
294:
282:
281:
269:
258:
256:
253:
252:
249:
244:
240:
229:
225:
216:
214:
210:
209:
206:
202:
197:
194:
190:
144:
141:
138:
103:
100:
87:
84:
40:logical matrix
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3141:
3130:
3127:
3125:
3122:
3120:
3119:Combinatorics
3117:
3115:
3112:
3111:
3109:
3094:
3091:
3089:
3086:
3084:
3083:
3078:
3072:
3071:
3068:
3062:
3059:
3057:
3054:
3052:
3051:Pseudoinverse
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3023:
3021:
3019:Related terms
3017:
3011:
3010:Z (chemistry)
3008:
3006:
3003:
3001:
2998:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2947:
2945:
2941:
2935:
2932:
2930:
2927:
2925:
2922:
2920:
2917:
2915:
2912:
2910:
2907:
2905:
2902:
2900:
2897:
2896:
2894:
2892:
2887:
2881:
2878:
2876:
2873:
2871:
2868:
2866:
2863:
2861:
2858:
2856:
2853:
2851:
2848:
2846:
2843:
2841:
2838:
2836:
2833:
2832:
2830:
2828:
2823:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2762:
2759:
2757:
2754:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
2689:
2687:
2684:
2682:
2679:
2677:
2674:
2672:
2669:
2667:
2664:
2663:
2661:
2657:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2597:
2594:
2592:
2589:
2588:
2586:
2584:
2580:
2575:
2569:
2566:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2539:
2536:
2534:
2531:
2530:
2528:
2526:
2521:
2515:
2512:
2510:
2507:
2505:
2502:
2500:
2497:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2475:
2472:
2470:
2467:
2466:
2464:
2460:
2454:
2451:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2409:
2406:
2404:
2401:
2399:
2396:
2394:
2391:
2389:
2386:
2384:
2381:
2379:
2376:
2374:
2371:
2369:
2368:Pentadiagonal
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2339:
2336:
2334:
2331:
2329:
2326:
2324:
2321:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2204:
2201:
2199:
2198:Anti-diagonal
2196:
2194:
2191:
2190:
2188:
2184:
2179:
2172:
2167:
2165:
2160:
2158:
2153:
2152:
2149:
2137:
2134:
2132:
2129:
2128:
2126:
2122:
2116:
2113:
2111:
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2092:
2090:
2086:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2030:
2027:
2025:
2022:
2020:
2017:
2016:
2014:
2012:
2008:
2000:
1997:
1996:
1994:
1990:
1987:
1986:
1985:Graph theory
1984:
1980:
1977:
1975:
1972:
1971:
1970:
1967:
1963:
1960:
1958:
1955:
1954:
1953:
1952:Combinatorics
1950:
1949:
1947:
1943:
1937:
1934:
1932:
1929:
1928:
1926:
1922:
1918:
1911:
1906:
1904:
1899:
1897:
1892:
1891:
1888:
1876:
1873:
1871:
1870:Graph drawing
1868:
1866:
1863:
1862:
1860:
1856:
1850:
1847:
1844:
1843:Newick format
1841:
1838:
1835:
1832:
1829:
1827:
1824:
1823:
1821:
1817:
1811:
1808:
1806:
1803:
1801:
1798:
1795:
1792:
1790:
1787:
1786:
1784:
1780:
1774:
1771:
1769:
1766:
1764:
1761:
1759:
1756:
1754:
1751:
1750:
1748:
1744:
1735:
1730:
1728:
1723:
1721:
1716:
1715:
1712:
1703:
1702:
1697:
1694:
1689:
1688:
1684:
1679:
1675:
1672:
1670:3-540-26183-4
1666:
1662:
1658:
1653:
1652:
1648:
1639:
1632:
1629:
1625:
1623:0-486-61480-8
1619:
1615:
1611:
1610:
1605:
1599:
1596:
1589:
1585:
1582:
1581:
1577:
1575:
1573:
1569:
1565:
1561:
1557:
1553:
1549:
1542:Block designs
1541:
1539:
1533:
1531:
1529:
1525:
1521:
1517:
1513:
1509:
1505:
1501:
1497:
1489:
1487:
1485:
1481:
1477:
1473:
1468:
1464:
1457:
1454:if the point
1450:
1446:
1442:
1437:
1433:
1429:
1425:
1421:
1416:
1412:
1407:
1404:
1400:
1392:
1390:
1388:
1380:
1374:
1361:
1356:
1350:
1345:
1340:
1335:
1328:
1323:
1318:
1313:
1306:
1301:
1296:
1291:
1284:
1279:
1274:
1269:
1263:
1253:
1250:
1243:
1240:
1237:
1234:
1231:
1230:
1226:
1223:
1220:
1217:
1214:
1213:
1209:
1206:
1203:
1200:
1197:
1196:
1192:
1189:
1186:
1183:
1180:
1179:
1172:
1166:
1160:
1154:
1152:
1151:
1148:
1147:
1144:
1137:
1130:
1128:
1126:
1122:
1114:
1112:
1109:
1105:
1097:
1095:
1093:
1089:
1085:
1081:
1077:
1073:
1070:The integral
1054:
1039:
1033:
1027:
1021:
1014:
1013:
1012:
1010:
1006:
1002:
999:The discrete
997:
995:
992:of dimension
991:
986:
982:
978:
974:
970:
966:
962:
958:
939:
934:
930:
926:
923:
917:
911:
896:
890:
887:
878:
872:
866:
859:
858:
857:
855:
851:
848:
844:
840:
835:
833:
829:
825:
821:
817:
813:
808:
777:
771:
761:
756:
752:
741:
737:
733:if edge
726:
720:
710:
705:
701:
690:
686:
682:if edge
674:
671:
663:
659:
654:
651:
647:
639:
638:
637:
635:
631:
627:
611:
608:
605:
597:
593:
588:
581:
568:
563:
557:
552:
547:
542:
535:
530:
525:
520:
513:
508:
503:
498:
491:
486:
481:
476:
470:
460:
457:
450:
447:
444:
441:
438:
437:
433:
430:
427:
424:
421:
420:
416:
413:
410:
407:
404:
403:
399:
396:
393:
390:
387:
386:
379:
373:
367:
361:
359:
358:
355:
354:
351:
335:
331:
327:
322:
318:
314:
309:
305:
301:
296:
292:
254:
247:
242:
238:
227:
223:
212:
204:
200:
195:
192:
188:
180:
179:
178:
176:
172:
168:
164:
160:
157:
142:
139:
136:
128:
124:
119:
117:
108:
101:
99:
97:
93:
85:
83:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
3073:
3005:Substitution
2918:
2891:graph theory
2388:Quaternionic
2378:Persymmetric
2124:Applications
1957:Block design
1930:
1837:LCF notation
1772:
1699:
1677:
1657:Graph Theory
1656:
1637:
1631:
1607:
1598:
1559:
1555:
1551:
1548:block design
1545:
1537:
1527:
1523:
1519:
1515:
1511:
1507:
1503:
1499:
1493:
1483:
1466:
1462:
1455:
1448:
1444:
1440:
1435:
1431:
1427:
1423:
1419:
1414:
1410:
1405:
1398:
1396:
1384:
1254:
1142:
1118:
1104:signed graph
1101:
1069:
1008:
1004:
998:
993:
984:
980:
976:
972:
968:
964:
960:
956:
954:
853:
849:
838:
836:
827:
823:
819:
811:
809:
806:
633:
629:
625:
591:
589:
586:
461:
283:
166:
162:
158:
126:
122:
120:
113:
92:graph theory
89:
86:Graph theory
79:
75:
71:
67:
63:
59:
55:
51:
47:
35:
29:
2980:Hamiltonian
2904:Biadjacency
2840:Correlation
2756:Householder
2706:Commutation
2443:Vandermonde
2438:Tridiagonal
2373:Permutation
2363:Nonnegative
2348:Matrix unit
2228:Bisymmetric
1995:Statistics
1875:Linked data
1381:Hypergraphs
1115:Multigraphs
1072:cycle space
816:orientation
125:(or simply
66:and column
32:mathematics
3108:Categories
2880:Transition
2875:Stochastic
2845:Covariance
2827:statistics
2806:Symplectic
2801:Similarity
2630:Unimodular
2625:Orthogonal
2610:Involutory
2605:Invertible
2600:Projection
2596:Idempotent
2538:Convergent
2433:Triangular
2383:Polynomial
2328:Hessenberg
2298:Equivalent
2293:Elementary
2273:Copositive
2263:Conference
2223:Bidiagonal
2024:Fano plane
1989:Hypergraph
1590:References
1484:vice versa
1480:hypergraph
1476:Levi graph
1387:hypergraph
1076:null space
847:line graph
784:otherwise.
261:otherwise.
3061:Wronskian
2985:Irregular
2975:Gell-Mann
2924:Laplacian
2919:Incidence
2899:Adjacency
2870:Precision
2835:Centering
2741:Generator
2711:Confusion
2696:Circulant
2676:Augmented
2635:Unipotent
2615:Nilpotent
2591:Congruent
2568:Stieltjes
2543:Defective
2533:Companion
2504:Redheffer
2423:Symmetric
2418:Sylvester
2393:Signature
2323:Hermitian
2303:Frobenius
2213:Arrowhead
2193:Alternant
1974:Incidence
1845:for trees
1763:Edge list
1701:MathWorld
1606:(1973) ,
1568:permanent
1534:Polytopes
1461:and line
1001:Laplacian
924:−
772:−
721:−
672:−
609:×
140:×
3124:Matrices
2889:Used in
2825:Used in
2786:Rotation
2761:Jacobian
2721:Distance
2701:Cofactor
2686:Carleman
2666:Adjugate
2650:Weighing
2583:inverses
2579:products
2548:Definite
2479:Identity
2469:Exchange
2462:Constant
2428:Toeplitz
2313:Hadamard
2283:Diagonal
2088:Theorems
1999:Blocking
1969:Geometry
1578:See also
1574:(SDRs).
1080:integers
171:vertices
161:, where
80:incident
70:is 1 if
2990:Overlap
2955:Density
2914:Edmonds
2791:Seifert
2751:Hessian
2716:Coxeter
2640:Unitary
2558:Hurwitz
2489:Of ones
2474:Hilbert
2408:Skyline
2353:Metzler
2343:Logical
2338:Integer
2248:Boolean
2180:classes
1800:GraphML
1614:166-167
1550:. Here
1474:of the
1418:matrix
988:is the
845:of its
624:matrix
2909:Degree
2850:Design
2781:Random
2771:Payoff
2766:Moment
2691:Cartan
2681:BĂ©zout
2620:Normal
2494:Pascal
2484:Lehmer
2413:Sparse
2333:Hollow
2318:Hankel
2253:Cauchy
2178:Matrix
1945:Fields
1667:
1620:
1432:points
1401:of an
955:where
628:where
156:matrix
2970:Gamma
2934:Tutte
2796:Shear
2509:Shift
2499:Pauli
2448:Walsh
2358:Moore
2238:Block
1833:(GML)
1810:XGMML
1793:DotML
1436:lines
1408:is a
1121:loops
1092:field
832:up to
598:is a
594:of a
175:edges
38:is a
34:, an
2776:Pick
2746:Gram
2514:Zero
2218:Band
2079:Dual
1796:GEXF
1789:DGML
1665:ISBN
1618:ISBN
1522:and
1434:and
1426:and
1397:The
1123:and
1084:real
810:The
632:and
590:The
173:and
165:and
121:The
74:and
2865:Hat
2598:or
2581:or
1826:DOT
1805:GXL
1452:= 1
1086:or
1082:or
350:):
30:In
3110::
1698:.
1659:,
1616:,
1413:Ă
1251:=
1244:6
1232:4
1227:6
1215:3
1210:0
1198:2
1193:0
1181:1
1094:.
996:.
971:,
458:=
451:1
439:4
434:1
422:3
417:0
405:2
400:0
388:1
2995:S
2453:Z
2170:e
2163:t
2156:v
1909:e
1902:t
1895:v
1733:e
1726:t
1719:v
1704:.
1560:X
1556:Y
1552:X
1528:e
1524:Y
1520:d
1516:X
1512:Y
1508:X
1504:Y
1500:X
1467:j
1463:L
1459:i
1456:p
1449:j
1447:,
1445:i
1441:B
1428:q
1424:p
1420:B
1415:q
1411:p
1406:C
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20:)
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