966:
979:
1084:
1061:
3951:
731:
954:
695:. The distance is the length of a shortest path connecting the vertices. Unless lengths of edges are explicitly provided, the length of a path is the number of edges in it. The distance matrix resembles a high power of the adjacency matrix, but instead of telling only whether or not two vertices are connected (i.e., the connection matrix, which contains
2356:
instead be scanned, which takes a larger amount of time, proportional to the number of vertices in the whole graph. On the other hand, testing whether there is an edge between two given vertices can be determined at once with an adjacency matrix, while requiring time proportional to the minimum degree of the two vertices with the adjacency list.
738:
712:
The convention followed here (for undirected graphs) is that each edge adds 1 to the appropriate cell in the matrix, and each loop (an edge from a vertex to itself) adds 2 to the appropriate cell on the diagonal in the matrix. This allows the degree of a vertex to be easily found by taking the sum of
187:
and graphs with loops by storing the number of edges between each two vertices in the corresponding matrix element, and by allowing nonzero diagonal elements. Loops may be counted either once (as a single edge) or twice (as two vertex-edge incidences), as long as a consistent convention is followed.
2355:
Besides the space tradeoff, the different data structures also facilitate different operations. Finding all vertices adjacent to a given vertex in an adjacency list is as simple as reading the list, and takes time proportional to the number of neighbors. With an adjacency matrix, an entire row must
2244:
divided by 3 or 6 depending on whether the graph is directed or not. We divide by those values to compensate for the overcounting of each triangle. In an undirected graph, each triangle will be counted twice for all three nodes, because the path can be followed clockwise or counterclockwise :
1044:
of a vertex can be computed by summing the entries of the corresponding column and the out-degree of vertex by summing the entries of the corresponding row. When using the second definition, the in-degree of a vertex is given by the corresponding row sum and the out-degree is given by the
2347:
An alternative form of adjacency matrix (which, however, requires a larger amount of space) replaces the numbers in each element of the matrix with pointers to edge objects (when edges are present) or null pointers (when there is no edge). It is also possible to store
1533:
1032:
The former definition is commonly used in graph theory and social network analysis (e.g., sociology, political science, economics, psychology). The latter is more common in other applied sciences (e.g., dynamical systems, physics, network science) where
2320: | / 16 bytes to represent an undirected graph. Although slightly more succinct representations are possible, this method gets close to the information-theoretic lower bound for the minimum number of bits needed to represent all
949:{\displaystyle {\begin{pmatrix}2&1&0&0&1&0\\1&0&1&0&1&0\\0&1&0&1&0&0\\0&0&1&0&1&1\\1&1&0&1&0&0\\0&0&0&1&0&0\end{pmatrix}}}
310:
1865:
1211:
2305:
without incurring the space overhead from storing the many zero entries in the adjacency matrix of the sparse graph. In the following section the adjacency matrix is assumed to be represented by an
1928:
1683:
1601:
1348:
1746:
2067:
2316: | / 8 bytes to represent a directed graph, or (by using a packed triangular format and only storing the lower triangular part of the matrix) approximately |
1559:(it is easy to check that it is an eigenvalue and it is the maximum because of the above bound). The multiplicity of this eigenvalue is the number of connected components of
1628:
1340:
1275:
1240:
1313:
218:
1782:
2488:. Algebraic Coding Theory and Information Theory: DIMACS Workshop, Algebraic Coding Theory and Information Theory. American Mathematical Society. p. 63.
2114:. These can therefore serve as isomorphism invariants of graphs. However, two graphs may possess the same set of eigenvalues but not be isomorphic. Such
1787:
3609:
2838:
2866:
1154:
188:
Undirected graphs often use the latter convention of counting loops twice, whereas directed graphs typically use the former convention.
2646:, p. 361: "There are two data structures that people often use to represent graphs, the adjacency list and the adjacency matrix."
998:
The adjacency matrix of a directed graph can be asymmetric. One can define the adjacency matrix of a directed graph either such that
3823:
3042:
3914:
2829:
2312:
Because each entry in the adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only |
2572:
2286:
in computer programs for manipulating graphs. The main alternative data structure, also in use for this application, is the
3833:
3599:
2301:
only store non-zero matrix entries and implicitly represent the zero entries. They can, for example, be used to represent
63:
2859:
2454:
2095:
1870:
175:, and zero when there is no edge. The diagonal elements of the matrix are all zero, since edges from a vertex to itself (
2752:
2293:
The space needed to represent an adjacency matrix and the time needed to perform operations on them is dependent on the
2689:
2609:
2493:
1633:
1528:{\displaystyle \lambda _{1}v_{x}=(Av)_{x}=\sum _{y=1}^{n}A_{x,y}v_{y}\leq \sum _{y=1}^{n}A_{x,y}v_{x}=v_{x}\deg(x).}
67:
2960:
965:
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3634:
2826:— an educational Java web start game demonstrating the relationship between adjacency matrices and graphs.
3997:
3181:
2852:
1711:
2340:, adjacency lists require less storage space, because they do not waste any space representing edges that are
2742:
978:
3398:
3035:
2707:
2298:
1243:
48:
36:
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2013:
1083:
2328:, fewer bits per byte can be used to ensure that all bytes are text characters, for instance by using a
3629:
3151:
2887:
2680:
2283:
2452:
Seidel, J. J. (1968). "Strongly
Regular Graphs with (−1, 1, 0) Adjacency Matrix Having Eigenvalue 3".
2394:, Cambridge Mathematical Library (2nd ed.), Cambridge University Press, Definition 2.1, p. 7
2230:. A great example of how this is useful is in counting the number of triangles in an undirected graph
1041:
3733:
3604:
3518:
2099:
3838:
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3436:
3116:
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1111:
contains all ones except along the diagonal where there are only zeros. The adjacency matrix of an
1606:
1318:
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1218:
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3802:
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3141:
3028:
643:
3743:
3326:
3131:
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180:
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to replace the nonzero elements with algebraic variables. The same concept can be extended to
3689:
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3106:
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102:
83:
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2841: : Application of the adjacency matrices to the computation generating series of walks.
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2728:
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2438:
2418:
2306:
305:{\displaystyle A={\begin{pmatrix}0_{r,r}&B\\B^{\mathsf {T}}&0_{s,s}\end{pmatrix}},}
40:
93:, a different matrix representation whose elements indicate whether vertex–edge pairs are
8:
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176:
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Open Data
Structures - Section 12.1 - AdjacencyMatrix: Representing a Graph by a Matrix
2001:
1767:
2245:
ijk or ikj. The adjacency matrix can be used to determine whether or not the graph is
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1997:
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Volume 68 of DIMACS series in discrete mathematics and theoretical computer science
2462:
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are taken to be the number of edges between the vertices or the weight of the edge
90:
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24:
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2115:
1761:
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1073:
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665:
417:
201:
1860:{\displaystyle \lambda (G)=\max _{\left|\lambda _{i}\right|<d}|\lambda _{i}|}
3919:
3863:
3843:
3828:
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3624:
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98:
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3111:
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representation. Besides avoiding wasted space, this compactness encourages
2309:
so that zero and non-zero entries are all directly represented in storage.
2302:
1749:
1206:{\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}.}
1069:
713:
the values in either its respective row or column in the adjacency matrix.
59:
55:
20:
1242:
is bounded above by the maximum degree. This can be seen as result of the
3649:
3619:
3387:
3221:
3091:
3009:
2119:
2107:
2103:
1144:
1137:
1116:
1112:
971:
375:
79:
2844:
3700:
3161:
1140:
535:
184:
75:
3934:
3508:
2897:
2833:
2814:
2325:
657:
2430:
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179:) are not allowed in simple graphs. It is also sometimes useful in
2705:
Turán, György (1984), "On the succinct representation of graphs",
3020:
2934:
89:
The adjacency matrix of a graph should be distinguished from its
2684:(Second ed.), MIT Press and McGraw-Hill, pp. 527–531,
2329:
1603:
for connected graphs. It can be shown that for each eigenvalue
2409:(1962), "The determinant of the adjacency matrix of a graph",
1285:
has maximum absolute value. Without loss of generality assume
2944:
2134:
is the adjacency matrix of the directed or undirected graph
1292:
is positive since otherwise you simply take the eigenvector
730:
2923:
2509:
Borgatti, Steve; Everett, Martin; Johnson, Jeffrey (2018),
2662:
382:
uniquely represents the graph, and the remaining parts of
2939:
2804:
1037:
is sometimes used to describe linear dynamics on graphs.
2190:
is the smallest nonnegative integer, such that for some
1151:
of the graph. It is common to denote the eigenvalues by
1091:
As the graph is directed, the matrix is not necessarily
2541:, Chapter 2 ("The spectrum of a graph"), pp. 7–13.
2508:
1132:
The adjacency matrix of an undirected simple graph is
747:
233:
2016:
1873:
1790:
1770:
1714:
1636:
1609:
1569:
1351:
1321:
1298:
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1221:
1157:
741:
221:
2528:(2nd ed.), Oxford University Press, p. 110
2678:(2001), "Section 22.1: Representations of graphs",
1923:{\displaystyle \lambda (G)\geq 2{\sqrt {d-1}}-o(1)}
652:-adjacency matrix. This matrix is used in studying
2559:, Universitext, New York: Springer, pp. 6–7,
2061:
1922:
1859:
1776:
1740:
1677:
1622:
1595:
1527:
1334:
1307:
1269:
1234:
1205:
948:
304:
16:Square matrix used to represent a graph or network
2352:directly in the elements of an adjacency matrix.
2156:) has an interesting interpretation: the element
986:White fields are zeros, colored fields are ones.
3979:
2777:
2643:
2551:Brouwer, Andries E.; Haemers, Willem H. (2012),
1807:
1147:basis. The set of eigenvalues of a graph is the
2550:
1678:{\displaystyle -\lambda _{i}=\lambda _{n+1-i}}
3036:
2860:
2839:Café math : Adjacency Matrices of Graphs
2170:gives the number of (directed or undirected)
1937:
699:), it gives the exact distance between them.
1596:{\displaystyle \lambda _{1}>\lambda _{2}}
74:. The relationship between a graph and the
70:are bidirectional), the adjacency matrix is
62:with zeros on its diagonal. If the graph is
2748:Description of graph6 and sparse6 encodings
3610:Fundamental (linear differential equation)
3043:
3029:
2867:
2853:
1942:Suppose two directed or undirected graphs
2874:
2621:
2479:
1934:, which have applications in many areas.
1741:{\displaystyle \lambda _{1}-\lambda _{2}}
157:is one when there is an edge from vertex
2480:Shum, Kenneth; Blake, Ian (2003-12-18).
2256:adjacency matrix (i.e., if there exists
3915:Matrix representation of conic sections
2658:
2656:
2654:
2652:
191:
101:, which contains information about the
3980:
2624:"Matrices with Permanent Equal to One"
2523:
2451:
2405:
2324:-vertex graphs. For storing graphs in
2278:The adjacency matrix may be used as a
1136:, and therefore has a complete set of
267:
82:of its adjacency matrix is studied in
3024:
2848:
2805:
2773:
2771:
2769:
2704:
2538:
2389:
1760:. It is also useful to introduce the
212:vertices can be written in the form
2649:
707:
2631:Linear Algebra and its Applications
2062:{\displaystyle PA_{1}P^{-1}=A_{2}.}
1246:, but it can be proved easily. Let
378:. In this case, the smaller matrix
136:, the adjacency matrix is a square
113:For a simple graph with vertex set
13:
3050:
2766:
2297:chosen for the underlying matrix.
2273:
2266:is the zero matrix), then it is a
993:
14:
4009:
2798:
2787:Algorithm Design and Applications
2741:
1250:be one eigenvector associated to
1102:
3949:
2513:(2nd ed.), SAGE, p. 20
2125:
1082:
1059:
1040:Using the first definition, the
977:
964:
729:
470:. The biadjacency matrix is the
54:In the special case of a finite
3817:Used in science and engineering
2755:from the original on 2001-04-30
2735:
2698:
2637:
2615:
2222:is the distance between vertex
386:can be discarded as redundant.
3060:Explicitly constrained entries
2644:Goodrich & Tamassia (2015)
2588:
2544:
2532:
2517:
2502:
2473:
2445:
2399:
2383:
2000:if and only if there exists a
1917:
1911:
1883:
1877:
1853:
1838:
1800:
1794:
1519:
1513:
1385:
1375:
681:the distance between vertices
1:
3834:Fundamental (computer vision)
2376:
2299:Sparse matrix representations
1930:. This bound is tight in the
1122:
576:
108:
2974:for cubic Hamiltonian graphs
2721:10.1016/0166-218X(84)90126-4
2708:Discrete Applied Mathematics
2622:Nicholson, Victor A (1975).
2467:10.1016/0024-3795(68)90008-6
2094:and therefore have the same
1867:. This number is bounded by
1623:{\displaystyle \lambda _{i}}
1335:{\displaystyle \lambda _{1}}
1270:{\displaystyle \lambda _{1}}
1235:{\displaystyle \lambda _{1}}
58:, the adjacency matrix is a
7:
3600:Duplication and elimination
3399:eigenvalues or eigenvectors
2482:"Expander graphs and codes"
2359:
1546:is the first eigenvalue of
1127:
702:
35:used to represent a finite
10:
4014:
3533:With specific applications
3162:Discrete Fourier Transform
2888:Graph (abstract data type)
2681:Introduction to Algorithms
2252:If a directed graph has a
1938:Isomorphism and invariants
1705:-regular bipartite graph.
1107:The adjacency matrix of a
1045:corresponding column sum.
1020:it indicates an edge from
43:indicate whether pairs of
3943:
3892:
3824:Cabibbo–Kobayashi–Maskawa
3816:
3762:
3698:
3532:
3451:Satisfying conditions on
3450:
3396:
3335:
3059:
2992:
2953:
2916:
2880:
2565:10.1007/978-1-4614-1939-6
2511:Analyzing Social Networks
2100:characteristic polynomial
1752:and it is related to the
1685:is also an eigenvalue of
2966:Graph Modelling Language
2553:"1.3.6 Bipartite graphs"
2284:representation of graphs
1960:with adjacency matrices
1701:is an eigenvalue of any
1244:Perron–Frobenius theorem
1215:The greatest eigenvalue
390:is sometimes called the
3182:Generalized permutation
2336:. However, for a large
2234:, which is exactly the
1281:the component in which
1009:indicates an edge from
654:strongly regular graphs
644:Seidel adjacency matrix
3988:Algebraic graph theory
3956:Mathematics portal
2602:Algebraic Graph Theory
2392:Algebraic Graph Theory
2390:Biggs, Norman (1993),
2371:Self-similarity matrix
2268:directed acyclic graph
2063:
1924:
1861:
1778:
1742:
1679:
1624:
1597:
1529:
1467:
1417:
1336:
1309:
1271:
1236:
1207:
1089:Coordinates are 0–23.
984:Coordinates are 0–23.
950:
603:of a simple graph has
306:
181:algebraic graph theory
150:such that its element
39:. The elements of the
3998:Graph data structures
2875:Graph representations
2668:Leiserson, Charles E.
2524:Newman, Mark (2018),
2334:locality of reference
2295:matrix representation
2064:
1925:
1862:
1779:
1743:
1680:
1625:
1598:
1530:
1447:
1397:
1337:
1315:, also associated to
1310:
1272:
1237:
1208:
958:Coordinates are 1–6.
951:
642:on the diagonal. The
307:
204:whose two parts have
196:The adjacency matrix
84:spectral graph theory
51:or not in the graph.
2984:Trivial Graph Format
2789:, Wiley, p. 363
2779:Goodrich, Michael T.
2307:array data structure
2014:
1871:
1788:
1768:
1712:
1634:
1607:
1567:
1349:
1319:
1296:
1254:
1219:
1155:
739:
542:, then the elements
219:
192:Of a bipartite graph
3905:Linear independence
3152:Diagonally dominant
2604:, Springer (2001),
2423:1962SIAMR...4..202H
1002:a non-zero element
3910:Matrix exponential
3900:Jordan normal form
3734:Fisher information
3605:Euclidean distance
3519:Totally unimodular
2954:Text-based formats
2810:"Adjacency matrix"
2807:Weisstein, Eric W.
2218:is positive, then
2138:, then the matrix
2096:minimal polynomial
2059:
2002:permutation matrix
1920:
1857:
1836:
1774:
1738:
1675:
1620:
1593:
1525:
1332:
1308:{\displaystyle -v}
1305:
1267:
1232:
1203:
1143:and an orthogonal
946:
940:
638:if it is not, and
599:-adjacency matrix
392:biadjacency matrix
302:
293:
3975:
3974:
3967:Category:Matrices
3839:Fuzzy associative
3729:Doubly stochastic
3437:Positive-definite
3117:Block tridiagonal
3018:
3017:
2917:XML-based formats
2783:Tamassia, Roberto
2672:Rivest, Ronald L.
2664:Cormen, Thomas H.
2574:978-1-4614-1938-9
2557:Spectra of Graphs
1903:
1806:
1777:{\displaystyle A}
1697:. In particular −
1542:-regular graphs,
1100:
1099:
1054:Adjacency matrix
991:
990:
724:Adjacency matrix
708:Undirected graphs
66:(i.e. all of its
4005:
3962:List of matrices
3954:
3953:
3930:Row echelon form
3874:State transition
3803:Seidel adjacency
3685:Totally positive
3545:Alternating sign
3142:Complex Hadamard
3045:
3038:
3031:
3022:
3021:
2993:Related concepts
2908:Incidence matrix
2903:Adjacency matrix
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2366:Laplacian matrix
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2116:linear operators
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2006:
1995:
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1968:
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1932:Ramanujan graphs
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1204:
1199:
1198:
1180:
1179:
1167:
1166:
1086:
1063:
1048:
1047:
1036:
1027:
1023:
1016:
1012:
1008:
981:
968:
955:
953:
952:
947:
945:
944:
733:
716:
715:
694:
687:
680:
669:has in position
651:
650:
641:
637:
633:
621:
602:
598:
597:
573:, respectively.
572:
550:
533:
526:
499:
483:
479:
469:
465:
442:
415:
389:
385:
381:
374:
364:
354:
341:
328:
318:
311:
309:
308:
303:
298:
297:
290:
289:
272:
271:
270:
251:
250:
211:
207:
199:
174:
165:
156:
149:
145:
135:
105:of each vertex.
97:or not, and its
91:incidence matrix
29:adjacency matrix
25:computer science
4013:
4012:
4008:
4007:
4006:
4004:
4003:
4002:
3978:
3977:
3976:
3971:
3948:
3939:
3888:
3812:
3758:
3694:
3528:
3446:
3392:
3331:
3132:Centrosymmetric
3055:
3049:
3019:
3014:
2988:
2949:
2912:
2881:Data structures
2876:
2873:
2801:
2796:
2795:
2776:
2767:
2758:
2756:
2740:
2736:
2703:
2699:
2692:
2676:Stein, Clifford
2661:
2650:
2642:
2638:
2626:
2620:
2616:
2593:
2589:
2575:
2549:
2545:
2537:
2533:
2522:
2518:
2507:
2503:
2496:
2478:
2474:
2455:Lin. Alg. Appl.
2450:
2446:
2431:10.1137/1004057
2404:
2400:
2388:
2384:
2379:
2362:
2321:
2276:
2274:Data structures
2261:
2257:
2239:
2231:
2227:
2223:
2219:
2213:
2200:
2199:
2195:
2191:
2187:
2183:
2179:
2175:
2158:
2157:
2153:
2149:
2139:
2135:
2131:
2128:
2118:are said to be
2088:
2082:
2079:
2073:
2072:In particular,
2050:
2046:
2034:
2030:
2024:
2020:
2015:
2012:
2011:
2004:
1994:
1988:
1985:
1979:
1976:
1970:
1967:
1961:
1958:
1952:
1949:
1943:
1940:
1892:
1872:
1869:
1868:
1852:
1846:
1842:
1837:
1819:
1815:
1811:
1810:
1789:
1786:
1785:
1769:
1766:
1765:
1762:spectral radius
1757:
1732:
1728:
1719:
1715:
1713:
1710:
1709:
1708:The difference
1702:
1698:
1695:bipartite graph
1690:
1686:
1657:
1653:
1644:
1640:
1635:
1632:
1631:
1630:, its opposite
1614:
1610:
1608:
1605:
1604:
1587:
1583:
1574:
1570:
1568:
1565:
1564:
1560:
1552:
1551:
1550:for the vector
1547:
1543:
1539:
1501:
1497:
1488:
1484:
1472:
1468:
1462:
1451:
1438:
1434:
1422:
1418:
1412:
1401:
1388:
1384:
1366:
1362:
1356:
1352:
1350:
1347:
1346:
1326:
1322:
1320:
1317:
1316:
1297:
1294:
1293:
1290:
1286:
1282:
1278:
1261:
1257:
1255:
1252:
1251:
1247:
1226:
1222:
1220:
1217:
1216:
1194:
1190:
1175:
1171:
1162:
1158:
1156:
1153:
1152:
1130:
1125:
1105:
1090:
1088:
1078:
1065:
1034:
1025:
1021:
1014:
1010:
1007:
1003:
996:
994:Directed graphs
985:
983:
970:
957:
939:
938:
933:
928:
923:
918:
913:
907:
906:
901:
896:
891:
886:
881:
875:
874:
869:
864:
859:
854:
849:
843:
842:
837:
832:
827:
822:
817:
811:
810:
805:
800:
795:
790:
785:
779:
778:
773:
768:
763:
758:
753:
743:
742:
740:
737:
736:
710:
705:
693:
689:
686:
682:
670:
666:distance matrix
648:
647:
639:
635:
623:
616:
604:
600:
583:
582:
579:
570:
561:
552:
548:
543:
534:is a bipartite
531:
521:
512:
503:
497:
485:
481:
475: ×
471:
467:
463:
454:
444:
440:
431:
421:
418:bipartite graph
398:
387:
383:
379:
370: ×
366:
360: ×
356:
353:
343:
340:
330:
324: ×
320:
316:
292:
291:
279:
275:
273:
266:
265:
261:
258:
257:
252:
240:
236:
229:
228:
220:
217:
216:
209:
205:
202:bipartite graph
197:
194:
173:
167:
164:
158:
155:
151:
147:
141: ×
137:
133:
124:
114:
111:
17:
12:
11:
5:
4011:
4001:
4000:
3995:
3990:
3973:
3972:
3970:
3969:
3964:
3959:
3944:
3941:
3940:
3938:
3937:
3932:
3927:
3922:
3920:Perfect matrix
3917:
3912:
3907:
3902:
3896:
3894:
3890:
3889:
3887:
3886:
3881:
3876:
3871:
3866:
3861:
3856:
3851:
3846:
3841:
3836:
3831:
3826:
3820:
3818:
3814:
3813:
3811:
3810:
3805:
3800:
3795:
3790:
3785:
3780:
3775:
3769:
3767:
3760:
3759:
3757:
3756:
3751:
3746:
3741:
3736:
3731:
3726:
3721:
3716:
3711:
3705:
3703:
3696:
3695:
3693:
3692:
3690:Transformation
3687:
3682:
3677:
3672:
3667:
3662:
3657:
3652:
3647:
3642:
3637:
3632:
3627:
3622:
3617:
3612:
3607:
3602:
3597:
3592:
3587:
3582:
3577:
3572:
3567:
3562:
3557:
3552:
3547:
3542:
3536:
3534:
3530:
3529:
3527:
3526:
3521:
3516:
3511:
3506:
3501:
3496:
3491:
3486:
3481:
3476:
3467:
3461:
3459:
3448:
3447:
3445:
3444:
3439:
3434:
3429:
3427:Diagonalizable
3424:
3419:
3414:
3409:
3403:
3401:
3397:Conditions on
3394:
3393:
3391:
3390:
3385:
3380:
3375:
3370:
3365:
3360:
3355:
3350:
3345:
3339:
3337:
3333:
3332:
3330:
3329:
3324:
3319:
3314:
3309:
3304:
3299:
3294:
3289:
3284:
3279:
3277:Skew-symmetric
3274:
3272:Skew-Hermitian
3269:
3264:
3259:
3254:
3249:
3244:
3239:
3234:
3229:
3224:
3219:
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3179:
3174:
3169:
3164:
3159:
3154:
3149:
3144:
3139:
3134:
3129:
3124:
3119:
3114:
3109:
3107:Block-diagonal
3104:
3099:
3094:
3089:
3084:
3082:Anti-symmetric
3079:
3077:Anti-Hermitian
3074:
3069:
3063:
3061:
3057:
3056:
3048:
3047:
3040:
3033:
3025:
3016:
3015:
3013:
3012:
3007:
3002:
3000:Graph database
2996:
2994:
2990:
2989:
2987:
2986:
2981:
2975:
2969:
2963:
2957:
2955:
2951:
2950:
2948:
2947:
2942:
2937:
2932:
2929:
2926:
2920:
2918:
2914:
2913:
2911:
2910:
2905:
2900:
2895:
2893:Adjacency list
2890:
2884:
2882:
2878:
2877:
2872:
2871:
2864:
2857:
2849:
2843:
2842:
2836:
2827:
2821:
2800:
2799:External links
2797:
2794:
2793:
2765:
2743:McKay, Brendan
2734:
2715:(3): 289–294,
2697:
2690:
2648:
2636:
2614:
2587:
2573:
2543:
2531:
2516:
2501:
2494:
2472:
2461:(2): 281–298.
2444:
2417:(3): 202–210,
2398:
2381:
2380:
2378:
2375:
2374:
2373:
2368:
2361:
2358:
2288:adjacency list
2280:data structure
2275:
2272:
2198:, the element
2146:matrix product
2127:
2124:
2086:
2077:
2070:
2069:
2058:
2053:
2049:
2045:
2040:
2037:
2033:
2027:
2023:
2019:
1992:
1983:
1974:
1965:
1956:
1947:
1939:
1936:
1919:
1916:
1913:
1910:
1907:
1902:
1899:
1896:
1891:
1888:
1885:
1882:
1879:
1876:
1855:
1849:
1845:
1840:
1834:
1831:
1827:
1822:
1818:
1814:
1809:
1805:
1802:
1799:
1796:
1793:
1773:
1748:is called the
1735:
1731:
1727:
1722:
1718:
1672:
1669:
1666:
1663:
1660:
1656:
1652:
1647:
1643:
1639:
1617:
1613:
1590:
1586:
1582:
1577:
1573:
1536:
1535:
1524:
1521:
1518:
1515:
1512:
1509:
1504:
1500:
1496:
1491:
1487:
1481:
1478:
1475:
1471:
1465:
1460:
1457:
1454:
1450:
1446:
1441:
1437:
1431:
1428:
1425:
1421:
1415:
1410:
1407:
1404:
1400:
1396:
1391:
1387:
1383:
1380:
1377:
1374:
1369:
1365:
1359:
1355:
1329:
1325:
1304:
1301:
1288:
1264:
1260:
1229:
1225:
1202:
1197:
1193:
1189:
1186:
1183:
1178:
1174:
1170:
1165:
1161:
1129:
1126:
1124:
1121:
1109:complete graph
1104:
1103:Trivial graphs
1101:
1098:
1097:
1080:
1076:
1056:
1055:
1052:
1051:Labeled graph
1030:
1029:
1018:
1005:
995:
992:
989:
988:
975:
961:
960:
943:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
908:
905:
902:
900:
897:
895:
892:
890:
887:
885:
882:
880:
877:
876:
873:
870:
868:
865:
863:
860:
858:
855:
853:
850:
848:
845:
844:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
812:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
780:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
748:
746:
734:
726:
725:
722:
709:
706:
704:
701:
697:Boolean values
691:
684:
608:
578:
575:
566:
557:
546:
540:weighted graph
517:
508:
501:if and only if
489:
459:
452:
436:
429:
397:Formally, let
355:represent the
345:
332:
313:
312:
301:
296:
288:
285:
282:
278:
274:
269:
264:
260:
259:
256:
253:
249:
246:
243:
239:
235:
234:
232:
227:
224:
193:
190:
171:
162:
153:
129:
122:
110:
107:
15:
9:
6:
4:
3:
2:
4010:
3999:
3996:
3994:
3991:
3989:
3986:
3985:
3983:
3968:
3965:
3963:
3960:
3958:
3957:
3952:
3946:
3945:
3942:
3936:
3933:
3931:
3928:
3926:
3925:Pseudoinverse
3923:
3921:
3918:
3916:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3897:
3895:
3893:Related terms
3891:
3885:
3884:Z (chemistry)
3882:
3880:
3877:
3875:
3872:
3870:
3867:
3865:
3862:
3860:
3857:
3855:
3852:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3832:
3830:
3827:
3825:
3822:
3821:
3819:
3815:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3770:
3768:
3766:
3761:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3710:
3707:
3706:
3704:
3702:
3697:
3691:
3688:
3686:
3683:
3681:
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3651:
3648:
3646:
3643:
3641:
3638:
3636:
3633:
3631:
3628:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3537:
3535:
3531:
3525:
3522:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3471:
3468:
3466:
3463:
3462:
3460:
3458:
3454:
3449:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3404:
3402:
3400:
3395:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3340:
3338:
3334:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3248:
3245:
3243:
3242:Pentadiagonal
3240:
3238:
3235:
3233:
3230:
3228:
3225:
3223:
3220:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3168:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3088:
3085:
3083:
3080:
3078:
3075:
3073:
3072:Anti-diagonal
3070:
3068:
3065:
3064:
3062:
3058:
3053:
3046:
3041:
3039:
3034:
3032:
3027:
3026:
3023:
3011:
3008:
3006:
3005:Graph drawing
3003:
3001:
2998:
2997:
2995:
2991:
2985:
2982:
2979:
2978:Newick format
2976:
2973:
2970:
2967:
2964:
2962:
2959:
2958:
2956:
2952:
2946:
2943:
2941:
2938:
2936:
2933:
2930:
2927:
2925:
2922:
2921:
2919:
2915:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2885:
2883:
2879:
2870:
2865:
2863:
2858:
2856:
2851:
2850:
2847:
2840:
2837:
2835:
2831:
2828:
2825:
2822:
2817:
2816:
2811:
2808:
2803:
2802:
2788:
2784:
2780:
2774:
2772:
2770:
2754:
2750:
2749:
2744:
2738:
2730:
2726:
2722:
2718:
2714:
2710:
2709:
2701:
2693:
2691:0-262-03293-7
2687:
2683:
2682:
2677:
2673:
2669:
2665:
2659:
2657:
2655:
2653:
2645:
2640:
2632:
2625:
2618:
2611:
2610:0-387-95241-1
2607:
2603:
2600:
2599:Royle, Gordon
2596:
2595:Godsil, Chris
2591:
2584:
2580:
2576:
2570:
2566:
2562:
2558:
2554:
2547:
2540:
2535:
2527:
2520:
2512:
2505:
2497:
2495:9780821871102
2491:
2487:
2483:
2476:
2468:
2464:
2460:
2457:
2456:
2448:
2440:
2436:
2432:
2428:
2424:
2420:
2416:
2412:
2408:
2407:Harary, Frank
2402:
2393:
2386:
2382:
2372:
2369:
2367:
2364:
2363:
2357:
2353:
2351:
2345:
2343:
2339:
2335:
2331:
2327:
2319:
2315:
2310:
2308:
2304:
2303:sparse graphs
2300:
2296:
2291:
2289:
2285:
2281:
2271:
2269:
2264:
2255:
2250:
2248:
2242:
2237:
2216:
2208:
2204:
2173:
2166:
2162:
2147:
2142:
2126:Matrix powers
2123:
2121:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2085:
2076:
2056:
2051:
2047:
2043:
2038:
2035:
2031:
2025:
2021:
2017:
2010:
2009:
2008:
2003:
1999:
1991:
1982:
1973:
1964:
1955:
1946:
1935:
1933:
1914:
1908:
1905:
1900:
1897:
1894:
1889:
1886:
1880:
1874:
1847:
1843:
1832:
1829:
1825:
1820:
1816:
1812:
1803:
1797:
1791:
1771:
1763:
1755:
1751:
1733:
1729:
1725:
1720:
1716:
1706:
1696:
1670:
1667:
1664:
1661:
1658:
1654:
1650:
1645:
1641:
1637:
1615:
1611:
1588:
1584:
1580:
1575:
1571:
1555:
1522:
1516:
1510:
1507:
1502:
1498:
1494:
1489:
1485:
1479:
1476:
1473:
1469:
1463:
1458:
1455:
1452:
1448:
1444:
1439:
1435:
1429:
1426:
1423:
1419:
1413:
1408:
1405:
1402:
1398:
1394:
1389:
1381:
1378:
1372:
1367:
1363:
1357:
1353:
1345:
1344:
1343:
1327:
1323:
1302:
1299:
1262:
1258:
1245:
1227:
1223:
1213:
1200:
1195:
1191:
1187:
1184:
1181:
1176:
1172:
1168:
1163:
1159:
1150:
1146:
1142:
1139:
1135:
1120:
1118:
1114:
1110:
1096:
1094:
1085:
1081:
1079:
1075:
1071:
1068:
1062:
1058:
1057:
1053:
1050:
1049:
1046:
1043:
1038:
1019:
1001:
1000:
999:
987:
980:
976:
974:
973:
967:
963:
962:
959:
941:
935:
930:
925:
920:
915:
910:
903:
898:
893:
888:
883:
878:
871:
866:
861:
856:
851:
846:
839:
834:
829:
824:
819:
814:
807:
802:
797:
792:
787:
782:
775:
770:
765:
760:
755:
750:
744:
735:
732:
728:
727:
723:
721:
720:Labeled graph
718:
717:
714:
700:
698:
678:
674:
668:
667:
661:
659:
655:
645:
631:
627:
620:
615:
611:
607:
595:
591:
587:
574:
569:
565:
560:
556:
549:
541:
537:
528:
525:
520:
516:
511:
507:
502:
496:
492:
488:
478:
474:
462:
458:
451:
447:
439:
435:
428:
424:
419:
413:
409:
405:
401:
395:
393:
377:
376:zero matrices
373:
369:
363:
359:
352:
348:
339:
335:
327:
323:
299:
294:
286:
283:
280:
276:
262:
254:
247:
244:
241:
237:
230:
225:
222:
215:
214:
213:
203:
189:
186:
182:
178:
170:
161:
144:
140:
132:
128:
121:
117:
106:
104:
100:
99:degree matrix
96:
92:
87:
85:
81:
77:
73:
69:
65:
61:
57:
52:
50:
46:
42:
38:
34:
33:square matrix
30:
26:
22:
3947:
3879:Substitution
3772:
3765:graph theory
3262:Quaternionic
3252:Persymmetric
2972:LCF notation
2902:
2813:
2786:
2757:, retrieved
2747:
2737:
2712:
2706:
2700:
2679:
2639:
2630:
2617:
2601:
2590:
2556:
2546:
2539:Biggs (1993)
2534:
2525:
2519:
2510:
2504:
2485:
2475:
2458:
2453:
2447:
2414:
2410:
2401:
2391:
2385:
2354:
2350:edge weights
2346:
2341:
2338:sparse graph
2317:
2313:
2311:
2292:
2277:
2262:
2251:
2240:
2214:
2206:
2202:
2178:from vertex
2164:
2160:
2140:
2129:
2083:
2074:
2071:
1989:
1980:
1971:
1962:
1953:
1944:
1941:
1750:spectral gap
1707:
1553:
1537:
1214:
1148:
1131:
1106:
1087:
1070:Cayley graph
1064:
1039:
1031:
997:
982:
969:
956:
711:
676:
672:
664:
662:
634:is an edge,
629:
625:
618:
613:
609:
605:
593:
589:
585:
580:
567:
563:
558:
554:
545:
529:
523:
518:
514:
509:
505:
494:
490:
486:
476:
472:
460:
456:
449:
445:
437:
433:
426:
422:
411:
407:
403:
399:
396:
391:
371:
367:
361:
357:
350:
346:
337:
333:
329:matrix, and
325:
321:
314:
195:
168:
159:
142:
138:
130:
126:
119:
115:
112:
88:
80:eigenvectors
60:(0,1)-matrix
56:simple graph
53:
28:
21:graph theory
18:
3854:Hamiltonian
3778:Biadjacency
3714:Correlation
3630:Householder
3580:Commutation
3317:Vandermonde
3312:Tridiagonal
3247:Permutation
3237:Nonnegative
3222:Matrix unit
3102:Bisymmetric
3010:Linked data
2824:Fluffschack
2411:SIAM Review
2226:and vertex
2144:(i.e., the
2120:isospectral
2108:determinant
2104:eigenvalues
1978:are given.
1784:denoted by
1556:= (1, …, 1)
1145:eigenvector
1141:eigenvalues
1117:zero matrix
1113:empty graph
972:Nauru graph
480:0–1 matrix
420:with parts
185:multigraphs
76:eigenvalues
3982:Categories
3754:Transition
3749:Stochastic
3719:Covariance
3701:statistics
3680:Symplectic
3675:Similarity
3504:Unimodular
3499:Orthogonal
3484:Involutory
3479:Invertible
3474:Projection
3470:Idempotent
3412:Convergent
3307:Triangular
3257:Polynomial
3202:Hessenberg
3172:Equivalent
3167:Elementary
3147:Copositive
3137:Conference
3097:Bidiagonal
2759:2012-02-10
2633:(12): 187.
2377:References
2326:text files
2260:such that
2182:to vertex
2174:of length
2152:copies of
2007:such that
1998:isomorphic
1123:Properties
1042:in-degrees
658:two-graphs
649:(−1, 1, 0)
577:Variations
536:multigraph
466:and edges
166:to vertex
109:Definition
64:undirected
3935:Wronskian
3859:Irregular
3849:Gell-Mann
3798:Laplacian
3793:Incidence
3773:Adjacency
3744:Precision
3709:Centering
3615:Generator
3585:Confusion
3570:Circulant
3550:Augmented
3509:Unipotent
3489:Nilpotent
3465:Congruent
3442:Stieltjes
3417:Defective
3407:Companion
3378:Redheffer
3297:Symmetric
3292:Sylvester
3267:Signature
3197:Hermitian
3177:Frobenius
3087:Arrowhead
3067:Alternant
2980:for trees
2898:Edge list
2834:Pat Morin
2815:MathWorld
2344:present.
2254:nilpotent
2247:connected
2036:−
1906:−
1898:−
1887:≥
1875:λ
1844:λ
1817:λ
1792:λ
1754:expansion
1730:λ
1726:−
1717:λ
1668:−
1655:λ
1642:λ
1638:−
1612:λ
1585:λ
1572:λ
1511:
1449:∑
1445:≤
1399:∑
1354:λ
1324:λ
1300:−
1259:λ
1224:λ
1192:λ
1188:≥
1185:⋯
1182:≥
1173:λ
1169:≥
1160:λ
1134:symmetric
1093:symmetric
484:in which
72:symmetric
3993:Matrices
3763:Used in
3699:Used in
3660:Rotation
3635:Jacobian
3595:Distance
3575:Cofactor
3560:Carleman
3540:Adjugate
3524:Weighing
3457:inverses
3453:products
3422:Definite
3353:Identity
3343:Exchange
3336:Constant
3302:Toeplitz
3187:Hadamard
3157:Diagonal
2785:(2015),
2753:archived
2526:Networks
2360:See also
2282:for the
1149:spectrum
1128:Spectrum
1067:Directed
703:Examples
95:incident
49:adjacent
45:vertices
3864:Overlap
3829:Density
3788:Edmonds
3665:Seifert
3625:Hessian
3590:Coxeter
3514:Unitary
3432:Hurwitz
3363:Of ones
3348:Hilbert
3282:Skyline
3227:Metzler
3217:Logical
3212:Integer
3122:Boolean
3054:classes
2935:GraphML
2729:0749658
2612:, p.164
2583:2882891
2439:0144330
2419:Bibcode
2092:similar
1342:. Then
455:, ...,
432:, ...,
146:matrix
3783:Degree
3724:Design
3655:Random
3645:Payoff
3640:Moment
3565:Cartan
3555:Bézout
3494:Normal
3368:Pascal
3358:Lehmer
3287:Sparse
3207:Hollow
3192:Hankel
3127:Cauchy
3052:Matrix
2727:
2688:
2608:
2581:
2571:
2492:
2437:
2330:Base64
319:is an
315:where
103:degree
41:matrix
3844:Gamma
3808:Tutte
3670:Shear
3383:Shift
3373:Pauli
3322:Walsh
3232:Moore
3112:Block
2968:(GML)
2945:XGMML
2928:DotML
2627:(PDF)
2236:trace
2186:. If
2172:walks
2112:trace
1693:is a
1115:is a
646:is a
416:be a
200:of a
177:loops
125:, …,
68:edges
37:graph
31:is a
27:, an
3650:Pick
3620:Gram
3388:Zero
3092:Band
2931:GEXF
2924:DGML
2686:ISBN
2606:ISBN
2569:ISBN
2490:ISBN
2110:and
2090:are
2081:and
1996:are
1987:and
1969:and
1951:and
1830:<
1581:>
1538:For
1277:and
1138:real
688:and
663:The
656:and
522:) ∈
365:and
342:and
208:and
78:and
47:are
23:and
3739:Hat
3472:or
3455:or
2961:DOT
2940:GXL
2717:doi
2561:doi
2463:doi
2427:doi
2342:not
2238:of
2212:of
2148:of
2130:If
1808:max
1764:of
1756:of
1689:if
1508:deg
1072:of
1024:to
1013:to
622:if
581:An
547:i,j
538:or
530:If
498:= 1
448:= {
425:= {
402:= (
118:= {
19:In
3984::
2832:,
2812:.
2781:;
2768:^
2751:,
2745:,
2725:MR
2723:,
2711:,
2674:;
2670:;
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2629:.
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2290:.
2270:.
2249:.
2205:,
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2122:.
2106:,
2102:,
2098:,
1119:.
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1017:or
1006:ij
675:,
660:.
628:,
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592:,
588:,
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527:.
513:,
443:,
410:,
406:,
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154:ij
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3869:S
3327:Z
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2044:=
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2005:P
1993:2
1990:G
1984:1
1981:G
1975:2
1972:A
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1957:2
1954:G
1948:1
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1554:v
1548:A
1544:d
1540:d
1523:.
1520:)
1517:x
1514:(
1503:x
1499:v
1495:=
1490:x
1486:v
1480:y
1477:,
1474:x
1470:A
1464:n
1459:1
1456:=
1453:y
1440:y
1436:v
1430:y
1427:,
1424:x
1420:A
1414:n
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1406:=
1403:y
1395:=
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1386:)
1382:v
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1376:(
1373:=
1368:x
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936:0
931:0
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766:0
761:0
756:1
751:2
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692:j
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400:G
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134:}
131:n
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