1826:
1709:
163:
3839:
674:
2581:
42:
2521:. However, for many questions it is better to treat vertices as indistinguishable. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges.) The same remarks apply to edges, so graphs with labeled edges are called
1724:
is a graph in which a number (the weight) is assigned to each edge. Such weights might represent for example costs, lengths or capacities, depending on the problem at hand. Such graphs arise in many contexts, for example in
1171:
863:
2804:
1327:
98:
The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person
1523:
1422:
2882:
has an underlying directed multigraph whose vertices are the objects of the category, and whose edges are the arrows of the category. In the language of category theory, one says that there is a
2288:. Cycle graphs can be characterized as connected graphs in which the degree of all vertices is 2. If a cycle graph occurs as a subgraph of another graph, it is a cycle or circuit in that graph.
2159:− 1. Path graphs can be characterized as connected graphs in which the degree of all but two vertices is 2 and the degree of the two remaining vertices is 1. If a path graph occurs as a
2649:
1781:
Some authors use "oriented graph" to mean the same as "directed graph". Some authors use "oriented graph" to mean any orientation of a given undirected graph or multigraph.
264:
3843:
354:
1442:
1240:
1347:
1208:
3183:
3214:"On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, – with three appendices"
1077:
775:
1190:
is an edge that joins a vertex to itself. Directed graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex
2654:
1860:
Most commonly in graph theory it is implied that the graphs discussed are finite. If the graphs are infinite, that is usually specifically stated.
1799:
is a graph in which each vertex has the same number of neighbours, i.e., every vertex has the same degree. A regular graph with vertices of degree
3016:(the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.
3485:
3790:
3758:
3739:
3716:
3697:
3678:
3659:
3640:
3619:
3600:
3550:
3423:
3375:
3278:
3137:
1245:
3813:
3217:
3102:
375:
is a generalization that allows multiple edges to have the same pair of endpoints. In some texts, multigraphs are simply called graphs.
625:
because an edge in a simple graph cannot start and end at the same vertex. Graphs with self-loops will be characterized by some or all
3399:
2909:
There are several operations that produce new graphs from initial ones, which might be classified into the following categories:
3143:
2981:
2517:
Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called
632:
being equal to a positive integer, and multigraphs (with multiple edges between vertices) will be characterized by some or all
3057:
1447:
1352:
984:
In one more general sense of the term allowing multiple edges, a directed graph is sometimes defined to be an ordered triple
3284:
65:
of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called
1963:
is a directed graph in which every ordered pair of vertices in the graph is strongly connected. Otherwise, it is called a
2887:
3509:
3097:
2963:
1909:
is an undirected graph in which every unordered pair of vertices in the graph is connected. Otherwise, it is called a
1869:
677:
A directed graph with three vertices and four directed edges (the double arrow represents an edge in each direction)
415:, because most results on finite graphs either do not extend to the infinite case or need a rather different proof.
3082:
1775:
1730:
2987:
2537:
may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.)
2485:
if the first one is the tail and the second one is the head of an edge), in which case the common edge is said to
2437:
3031:
2975:
2969:
2957:
2460:
2336:
2160:
2049:
1976:
463:
150:
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related
3179:
1837:
is a graph in which each pair of vertices is joined by an edge. A complete graph contains all possible edges.
2429:
494:
of a vertex is the number of edges that are incident to it; for graphs with loops, a loop is counted twice.
981:, not allowed under the definition above, are two or more edges with both the same tail and the same head.
2529:. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called
2447:
2443:
2370:
1982:
320:
on them. A vertex may belong to no edge, in which case it is not joined to any other vertex and is called
151:
135:
32:
3878:
3046:
2307:
1726:
67:
3827:
2592:
1952:
after replacing all of its directed edges with undirected edges. Otherwise, the ordered pair is called
134:
Graphs are the basic subject studied by graph theory. The word "graph" was first used in this sense by
517:
if loops are allowed, because a loop contributes 2 to the degree), and the maximum number of edges is
3050:
2891:
2869:
2817:
2433:
2321:
2185:
is a graph whose vertices and edges can be drawn in a plane such that no two of the edges intersect.
1535:
223:
50:
27:
This article is about sets of vertices connected by edges. For graphs of mathematical functions, see
1329:. So to allow loops the definitions must be expanded. For directed simple graphs, the definition of
3463:
2314:
2297:
2164:
1614:
is a graph in which some edges may be directed and some may be undirected. It is an ordered triple
1186:
380:
28:
1210:
to itself is the edge (for a directed simple graph) or is incident on (for a directed multigraph)
3782:
3240:
3061:
3009:
2425:
2022:
1829:
A complete graph with five vertices and ten edges. Each vertex has an edge to every other vertex.
543:
139:
2477:
if the head of the first one is the tail of the second one. Similarly, two vertices are called
3851:
3786:
3754:
3735:
3712:
3693:
3674:
3655:
3636:
3615:
3596:
3546:
3419:
3395:
3371:
3274:
3133:
2883:
1033:
1011:
741:
719:
92:
62:
3270:
3213:
1967:
if every ordered pair of vertices in the graph is weakly connected. Otherwise it is called a
327:
3521:
3475:
3232:
3198:
3077:
3039:
2937:
2919:
2904:
2813:
2809:
2042:
640:
575:
1427:
1213:
3513:
3221:
3187:
3035:
2943:
2875:
2824:
2541:
2318:
2017:
2010:
1547:
412:
384:, which are edges that join a vertex to itself. To allow loops, the pairs of vertices in
3728:
3107:
3087:
2879:
2545:
2464:
2400:
1820:
1332:
1193:
977:
668:
408:
95:
as a set of dots or circles for the vertices, joined by lines or curves for the edges.
3505:
Pankaj Gupta, Ashish Goel, Jimmy Lin, Aneesh Sharma, Dong Wang, and Reza Bosagh Zadeh
2514:, but the terminology is not consistent and not all mathematicians allow this object.
3872:
3776:
3127:
3092:
2418:
2406:
1790:
2525:. Graphs with labels attached to edges or vertices are more generally designated as
1774:
may be edges of the graph. That is, it is a directed graph that can be formed as an
131:, then this graph is directed, because owing money is not necessarily reciprocated.
17:
3065:
3027:
2865:
2176:
1993:
vertices (respectively, edges) exists that, when removed, disconnects the graph. A
1071:
769:
191:
54:
3480:
3572:
3260:
3034:. But in that case, there is no limitation on the number of edges: it can be any
1166:{\displaystyle \phi :E\to \{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\}}
858:{\displaystyle E\subseteq \{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\}}
388:
must be allowed to have the same node twice. Such generalized graphs are called
3464:"A social network analysis of Twitter: Mapping the digital humanities community"
3132:(Corrected, enlarged republication. ed.). New York: Dover Pub. p. 19.
2194:
1825:
1605:
419:
3053:
introduces power graphs as an alternative representation of undirected graphs.
1708:
3205:. From page 284: "Every invariant and covariant thus becomes expressible by a
3002:
2931:
2925:
2799:{\displaystyle E=\{\{1,2\},\{1,5\},\{2,3\},\{2,5\},\{3,4\},\{4,5\},\{4,6\}\}.}
2506:
2085:
1850:
372:
187:
2388:) is a directed acyclic graph whose underlying undirected graph is a forest.
615:
is either 0, indicating disconnection, or 1, indicating connection; moreover
3859:
3630:
3525:
3506:
3266:
423:
3854:
3838:
3418:(International student ed.). Boston: PWS-KENT Pub. Co. p. 463.
2348:
1074:
of vertices (that is, an edge is associated with two distinct vertices):
3561:
2473:
if they share a common vertex. Two edges of a directed graph are called
3244:
3020:
3013:
2858:
2557:
2412:
1525:. To avoid ambiguity, these types of objects may be called precisely a
162:
1322:{\displaystyle \{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\}}
3202:
2589:
The diagram is a schematic representation of the graph with vertices
2335:
path, or equivalently an acyclic undirected graph, or equivalently a
3236:
1700:
defined as above. Directed and undirected graphs are special cases.
947:. A vertex may exist in a graph and not belong to an edge. The edge
673:
2580:
41:
3518:
Proceedings of the 22nd international conference on World Wide Web
2864:
Particularly regular examples of directed graphs are given by the
2579:
2504:. The graph with no vertices and no edges is sometimes called the
2331:
is an undirected graph in which any two vertices are connected by
1707:
1177:
To avoid ambiguity, this type of object may be called precisely a
869:
To avoid ambiguity, this type of object may be called precisely a
672:
161:
40:
3349:
3347:
639:
being equal to a positive integer. Undirected graphs will have a
411:
are considered, but they are usually viewed as a special kind of
3574:
Lists, Decisions and Graphs. With an
Introduction to Probability
3064:
are closely modeled after graphs, and borrow many concepts from
3068:
to perform spatial analysis on road networks or utility grids.
2489:
the two vertices. An edge and a vertex on that edge are called
3209:
precisely identical with a Kekuléan diagram or chemicograph."
2954:, which create a new graph from two initial ones, such as:
2214:
is a graph in which the vertices can be listed in an order
2105:
is a graph in which the vertices can be listed in an order
3247:. The term "graph" first appears in this paper on page 65.
2496:
The graph with only one vertex and no edges is called the
396:
when it is clear from the context that loops are allowed.
3781:(Corrected, enlarged republication. ed.). New York:
2916:, which create a new graph from an initial one, such as:
2812:, directed graphs are used to represent knowledge (e.g.,
2500:. A graph with only vertices and no edges is known as an
2041:
share a common edge. Alternatively, it is a graph with a
138:
in 1878 due to a direct relation between mathematics and
79:) and each of the related pairs of vertices is called an
3414:
Fletcher, Peter; Hoyle, Hughes; Patty, C. Wayne (1991).
2857:
A directed graph can model information networks such as
1849:
is a graph in which the vertex set and the edge set are
462:. However, in some contexts, such as for expressing the
3156:
A graph is an object consisting of two sets called its
1750:
is that it is a directed graph in which at most one of
1518:{\displaystyle \phi :E\to \{(x,y)\mid (x,y)\in V^{2}\}}
1417:{\displaystyle E\subseteq \{(x,y)\mid (x,y)\in V^{2}\}}
692:
In one restricted but very common sense of the term, a
403:
is taken to be finite (which implies that the edge set
1874:
In an undirected graph, an unordered pair of vertices
1542:
The edges of a directed simple graph permitting loops
486:(otherwise, a non-empty graph could have size 0). The
2657:
2595:
1450:
1430:
1355:
1335:
1248:
1216:
1196:
1080:
778:
330:
226:
3005:, an edge can join any positive number of vertices.
2052:, the vertex set is the union of two disjoint sets,
3635:(3rd ed.). Berlin, New York: Springer-Verlag.
2373:(DAG) whose underlying undirected graph is a tree.
1712:
A weighted graph with ten vertices and twelve edges
3727:
2798:
2643:
1517:
1436:
1416:
1341:
1321:
1234:
1202:
1165:
857:
426:of vertices (and thus an empty set of edges). The
348:
258:
3225:American Journal of Mathematics, Pure and Applied
2021:is a simple graph in which the vertex set can be
1997:-vertex-connected graph is often called simply a
1916:In a directed graph, an ordered pair of vertices
598:specifying the number of connections from vertex
3650:Graham, R.L.; Grötschel, M.; Lovász, L. (1995).
3353:
3329:
3304:
3571:Bender, Edward A.; Williamson, S. Gill (2010).
574:is an edge. A graph is fully determined by its
1424:. For directed multigraphs, the definition of
3751:CRC Standard Mathematical Tables and Formulae
3563:Digraphs: Theory, Algorithms and Applications
689:is a graph in which edges have orientations.
8:
3726:Iyanaga, Shôkichi; Kawada, Yukiyosi (1977).
3595:(2nd ed.). Cambridge University Press.
2790:
2787:
2775:
2769:
2757:
2751:
2739:
2733:
2721:
2715:
2703:
2697:
2685:
2679:
2667:
2664:
2638:
2602:
1512:
1463:
1411:
1362:
1316:
1249:
1160:
1093:
852:
785:
343:
331:
253:
227:
142:(what he called a chemico-graphical image).
2037:share a common edge and no two vertices in
166:A graph with three vertices and three edges
1898:. Otherwise, the unordered pair is called
1807:‑regular graph or regular graph of degree
1306:
1298:
1150:
1142:
842:
834:
3753:(31st ed.). Chapman & Hall/CRC.
3479:
2868:of finitely-generated groups, as well as
2656:
2594:
2584:A graph with six vertices and seven edges
1506:
1449:
1429:
1405:
1354:
1334:
1300:
1299:
1292:
1247:
1215:
1195:
1144:
1143:
1136:
1079:
836:
835:
828:
777:
378:Sometimes, graphs are allowed to contain
329:
247:
234:
225:
45:A graph with six vertices and seven edges
3688:Gross, Jonathan L.; Yellen, Jay (2003).
3669:Gross, Jonathan L.; Yellen, Jay (1998).
3507:WTF: The who-to-follow system at Twitter
3259:Gross, Jonathan L.; Yellen, Jay (2004).
2306:is an undirected graph in which any two
1940:. Otherwise, the ordered pair is called
1824:
115:. In contrast, if an edge from a person
3584:Théorie des graphes et ses applications
3316:See, for instance, Iyanaga and Kawada,
3118:
503:, the maximum degree of each vertex is
266:of vertices, whose elements are called
3730:Encyclopedic Dictionary of Mathematics
3394:(4th ed.), Pearson, p. 405,
2842:is a direct predecessor of an element
1527:directed simple graph permitting loops
3711:. Addison Wesley Publishing Company.
3008:An undirected graph can be seen as a
2834:defines a directed graph. An element
2820:, and many other discrete structures.
91:). Typically, a graph is depicted in
7:
3103:List of publications in graph theory
1531:directed multigraph permitting loops
38:Vertices connected in pairs by edges
3560:Bang-Jensen, J.; Gutin, G. (2000).
3416:Foundations of Discrete Mathematics
3368:Linear Algebra and Its Applications
3178:J. J. Sylvester (February 7, 1878)
2396:More advanced kinds of graphs are:
212:is a set whose elements are called
179:
2861:, with one user following another.
1778:of an undirected (simple) graph.
25:
3671:Graph Theory and Its Applications
2644:{\displaystyle V=\{1,2,3,4,5,6\}}
1944:if an undirected path leads from
1586:to one another, which is denoted
924:of the edge. The edge is said to
3837:
2469:Two edges of a graph are called
440:of vertices, usually denoted by
3488:from the original on 2021-03-02
3287:from the original on 2023-02-04
3146:from the original on 5 May 2019
2982:lexicographic product of graphs
2533:. (In the literature, the term
2064:is adjacent to every vertex in
458:of edges, typically denoted by
259:{\displaystyle \{v_{1},v_{2}\}}
61:is a structure consisting of a
3058:geographic information systems
2068:but there are no edges within
1986:is a graph in which no set of
1932:if a directed path leads from
1562:. Specifically, for each edge
1496:
1484:
1478:
1466:
1460:
1395:
1383:
1377:
1365:
1282:
1270:
1264:
1252:
1229:
1217:
1126:
1114:
1108:
1096:
1090:
818:
806:
800:
788:
542:The edges of a graph define a
103:can shake hands with a person
1:
3545:(1st ed.). McGraw-Hill.
3481:10.1080/23311983.2016.1171458
3434:is a graph in which a number
3370:(4th ed.), Brooks Cole,
2481:if they share a common edge (
2033:, so that no two vertices in
1853:. Otherwise, it is called an
550:. Specifically, two vertices
3844:Graph (discrete mathematics)
3778:Introduction to Graph Theory
3775:Trudeau, Richard J. (1993).
3541:Balakrishnan, V. K. (1997).
3468:Cogent Arts & Humanities
3354:Bender & Williamson 2010
3330:Bender & Williamson 2010
3305:Bender & Williamson 2010
3129:Introduction to Graph Theory
3126:Trudeau, Richard J. (1993).
3019:Every graph gives rise to a
2888:category of small categories
2237:such that the edges are the
2128:such that the edges are the
546:on the vertices, called the
220:is a set of unordered pairs
3749:Zwillinger, Daniel (2002).
3446:, is assigned to each edge
3098:List of graph theory topics
2964:cartesian product of graphs
1870:Connectivity (graph theory)
407:is also finite). Sometimes
3895:
3629:Diestel, Reinhard (2005).
3614:(1st ed.). Springer.
3586:(in French). Paris: Dunod.
3462:Grandjean, Martin (2016).
3083:Graph (abstract data type)
2902:
2458:
2446:and their generalizations
2438:distance-transitive graphs
2346:
2295:
2192:
2174:
2163:of another graph, it is a
2083:
2060:, so that every vertex in
2008:
1867:
1818:
1788:
1731:traveling salesman problem
1603:
666:
643:adjacency matrix (meaning
399:Generally, the vertex set
26:
3828:Resources in your library
3652:Handbook of Combinatorics
1070:mapping every edge to an
470:is used for the quantity
448:of a graph is its number
430:of a graph is its number
186:to distinguish it from a
178:to distinguish it from a
3690:Handbook of Graph Theory
3392:Java Software Structures
3366:Strang, Gilbert (2005),
3320:, p. 234 or Biggs, p. 4.
3262:Handbook of graph theory
2976:strong product of graphs
2970:tensor product of graphs
2958:disjoint union of graphs
2461:Glossary of graph theory
2424:other graphs with large
2403:and its generalizations;
2367:singly connected network
2050:complete bipartite graph
1977:k-vertex-connected graph
1961:strongly connected graph
1682:(the undirected edges),
466:of algorithms, the term
464:computational complexity
216:(singular: vertex), and
3610:Bollobás, Béla (2002).
3526:10.1145/2488388.2488433
3212:J. J. Sylvester (1878)
3180:"Chemistry and algebra"
2448:distance-regular graphs
2444:strongly regular graphs
539:if loops are allowed).
422:is a graph that has an
349:{\displaystyle \{u,v\}}
152:mathematical structures
111:also shakes hands with
3707:Harary, Frank (1995).
3593:Algebraic Graph Theory
3591:Biggs, Norman (1993).
3582:Berge, Claude (1958).
2988:series–parallel graphs
2800:
2645:
2585:
2371:directed acyclic graph
1983:k-edge-connected graph
1965:weakly connected graph
1830:
1727:shortest path problems
1713:
1686:(the directed edges),
1519:
1444:should be modified to
1438:
1418:
1349:should be modified to
1343:
1323:
1236:
1204:
1167:
859:
772:of distinct vertices:
678:
606:. For a simple graph,
350:
305:. The edge is said to
301:are called the edge's
260:
167:
46:
33:Graph (disambiguation)
31:. For other uses, see
3047:computational biology
2870:Schreier coset graphs
2818:finite state machines
2801:
2646:
2583:
2544:of all graphs is the
1890:if a path leads from
1828:
1746:One definition of an
1711:
1550:~ on the vertices of
1520:
1439:
1437:{\displaystyle \phi }
1419:
1344:
1324:
1237:
1235:{\displaystyle (x,x)}
1205:
1168:
871:directed simple graph
860:
676:
356:exists, the vertices
351:
261:
174:(sometimes called an
165:
44:
3846:at Wikimedia Commons
3390:Lewis, John (2013),
3341:Graham et al., p. 5.
3051:power graph analysis
3030:, a graph is just a
2655:
2593:
2455:Properties of graphs
2317:, or equivalently a
1548:homogeneous relation
1448:
1428:
1353:
1333:
1246:
1214:
1194:
1078:
776:
591:square matrix, with
497:In a graph of order
328:
224:
51:discrete mathematics
18:Graph (graph theory)
3819:Graph (mathematics)
3612:Modern Graph Theory
2892:category of quivers
2556:: Set → Set is the
2426:automorphism groups
2298:Tree (graph theory)
2268:− 1, plus the edge
1554:that is called the
1179:directed multigraph
29:Graph of a function
3852:Weisstein, Eric W.
3783:Dover Publications
3512:2019-07-12 at the
3231:(1) : 64–90.
3220:2023-02-04 at the
3186:2023-02-04 at the
3062:geometric networks
3010:simplicial complex
2796:
2641:
2586:
2324:undirected graph.
1969:disconnected graph
1930:strongly connected
1911:disconnected graph
1831:
1714:
1635:mixed simple graph
1556:adjacency relation
1515:
1434:
1414:
1339:
1319:
1232:
1200:
1163:
1068:incidence function
855:
679:
548:adjacency relation
544:symmetric relation
346:
256:
168:
140:chemical structure
53:, particularly in
47:
3842:Media related to
3814:Library resources
3792:978-0-486-67870-2
3760:978-1-58488-291-6
3741:978-0-262-09016-2
3718:978-0-201-41033-4
3699:978-1-58488-090-5
3680:978-0-8493-3982-0
3661:978-0-262-07169-7
3642:978-3-540-26183-4
3621:978-0-387-98488-9
3602:978-0-521-45897-9
3552:978-0-07-005489-9
3425:978-0-53492-373-0
3377:978-0-03-010567-8
3280:978-1-58488-090-5
3139:978-0-486-67870-2
2952:binary operations
2884:forgetful functor
2430:vertex-transitive
2310:are connected by
1999:k-connected graph
1342:{\displaystyle E}
1303:
1203:{\displaystyle x}
1147:
839:
390:graphs with loops
93:diagrammatic form
16:(Redirected from
3886:
3865:
3864:
3841:
3803:
3801:
3799:
3764:
3745:
3733:
3722:
3703:
3684:
3665:
3646:
3625:
3606:
3587:
3578:
3567:
3556:
3529:
3503:
3497:
3496:
3494:
3493:
3483:
3459:
3453:
3452:
3411:
3405:
3404:
3387:
3381:
3380:
3363:
3357:
3351:
3342:
3339:
3333:
3327:
3321:
3314:
3308:
3302:
3296:
3295:
3293:
3292:
3256:
3250:
3203:10.1038/017284a0
3173:
3167:
3166:
3153:
3151:
3123:
3078:Conceptual graph
3040:continuous graph
3012:consisting of 1-
2938:complement graph
2920:edge contraction
2914:unary operations
2905:Graph operations
2899:Graph operations
2814:conceptual graph
2810:computer science
2805:
2803:
2802:
2797:
2650:
2648:
2647:
2642:
2392:Advanced classes
2287:
2259:
2213:
2150:
2104:
2043:chromatic number
1992:
1942:weakly connected
1927:
1885:
1773:
1761:
1699:
1692:
1685:
1681:
1677:
1672:mixed multigraph
1669:
1632:
1595:
1581:
1577:
1574:, its endpoints
1573:
1561:
1553:
1545:
1539:) respectively.
1524:
1522:
1521:
1516:
1511:
1510:
1443:
1441:
1440:
1435:
1423:
1421:
1420:
1415:
1410:
1409:
1348:
1346:
1345:
1340:
1328:
1326:
1325:
1320:
1305:
1304:
1301:
1297:
1296:
1242:which is not in
1241:
1239:
1238:
1233:
1209:
1207:
1206:
1201:
1172:
1170:
1169:
1164:
1149:
1148:
1145:
1141:
1140:
1065:
1031:
1009:
1002:
974:
958:
946:
942:
934:
930:
919:
916:of the edge and
911:
903:
899:
895:
891:
887:
864:
862:
861:
856:
841:
840:
837:
833:
832:
739:
717:
710:
658:
638:
631:
624:
614:
605:
601:
597:
590:
580:
576:adjacency matrix
573:
557:
553:
538:
527:
516:
509:
502:
485:
483:
477:
461:
457:
455:
443:
439:
437:
406:
402:
387:
363:
359:
355:
353:
352:
347:
315:
311:
300:
288:
284:
265:
263:
262:
257:
252:
251:
239:
238:
219:
211:
207:
176:undirected graph
21:
3894:
3893:
3889:
3888:
3887:
3885:
3884:
3883:
3869:
3868:
3850:
3849:
3834:
3833:
3832:
3822:
3821:
3817:
3810:
3797:
3795:
3793:
3774:
3771:
3769:Further reading
3761:
3748:
3742:
3725:
3719:
3706:
3700:
3687:
3681:
3668:
3662:
3649:
3643:
3628:
3622:
3609:
3603:
3590:
3581:
3570:
3559:
3553:
3540:
3537:
3532:
3514:Wayback Machine
3504:
3500:
3491:
3489:
3461:
3460:
3456:
3426:
3413:
3412:
3408:
3402:
3389:
3388:
3384:
3378:
3365:
3364:
3360:
3352:
3345:
3340:
3336:
3328:
3324:
3315:
3311:
3303:
3299:
3290:
3288:
3281:
3258:
3257:
3253:
3237:10.2307/2369436
3222:Wayback Machine
3188:Wayback Machine
3174:
3170:
3149:
3147:
3140:
3125:
3124:
3120:
3116:
3074:
3036:cardinal number
2999:
2997:Generalizations
2944:graph rewriting
2907:
2901:
2876:category theory
2850:if and only if
2825:binary relation
2653:
2652:
2591:
2590:
2578:
2467:
2457:
2394:
2386:oriented forest
2382:directed forest
2351:
2345:
2300:
2294:
2285:
2278:
2269:
2257:
2247:
2238:
2236:
2227:
2220:
2208:
2197:
2191:
2179:
2173:
2167:in that graph.
2148:
2138:
2129:
2127:
2118:
2111:
2099:
2088:
2082:
2025:into two sets,
2018:bipartite graph
2013:
2011:Bipartite graph
2007:
2005:Bipartite graph
1987:
1917:
1907:connected graph
1875:
1872:
1866:
1864:Connected graph
1843:
1823:
1817:
1793:
1787:
1763:
1751:
1744:
1739:
1737:Types of graphs
1706:
1698:
1694:
1691:
1687:
1683:
1679:
1675:
1666:
1659:
1638:
1615:
1608:
1602:
1587:
1582:are said to be
1579:
1575:
1563:
1559:
1551:
1543:
1502:
1446:
1445:
1426:
1425:
1401:
1351:
1350:
1331:
1330:
1288:
1244:
1243:
1212:
1211:
1192:
1191:
1132:
1076:
1075:
1063:
1029:
1007:
985:
964:
948:
944:
940:
932:
928:
917:
909:
904:are called the
901:
897:
896:, the vertices
893:
889:
877:
824:
774:
773:
737:
715:
697:
671:
665:
656:
649:
644:
637:
633:
630:
626:
621:
616:
612:
607:
603:
599:
596:
592:
582:
578:
563:
555:
551:
529:
518:
511:
504:
498:
479:
478:| + |
473:
471:
459:
451:
449:
441:
433:
431:
413:binary relation
409:infinite graphs
404:
400:
385:
361:
357:
326:
325:
324:. When an edge
313:
309:
290:
286:
282:
243:
230:
222:
221:
217:
209:
194:
160:
148:
136:J. J. Sylvester
39:
36:
23:
22:
15:
12:
11:
5:
3892:
3890:
3882:
3881:
3871:
3870:
3867:
3866:
3847:
3831:
3830:
3824:
3823:
3812:
3811:
3809:
3808:External links
3806:
3805:
3804:
3791:
3770:
3767:
3766:
3765:
3759:
3746:
3740:
3723:
3717:
3704:
3698:
3685:
3679:
3666:
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3641:
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3601:
3588:
3579:
3568:
3557:
3551:
3536:
3533:
3531:
3530:
3498:
3474:(1): 1171458.
3454:
3442:), called its
3432:weighted graph
3424:
3406:
3401:978-0133250121
3400:
3382:
3376:
3358:
3356:, p. 161.
3343:
3334:
3332:, p. 149.
3322:
3309:
3307:, p. 148.
3297:
3279:
3251:
3249:
3248:
3210:
3168:
3138:
3117:
3115:
3112:
3111:
3110:
3108:Network theory
3105:
3100:
3095:
3090:
3088:Graph database
3085:
3080:
3073:
3070:
2998:
2995:
2994:
2993:
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2985:
2979:
2973:
2967:
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2949:
2948:
2947:
2941:
2935:
2929:
2923:
2903:Main article:
2900:
2897:
2896:
2895:
2880:small category
2872:
2862:
2855:
2821:
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2622:
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2613:
2610:
2607:
2604:
2601:
2598:
2577:
2574:
2546:comma category
2519:vertex-labeled
2502:edgeless graph
2465:Graph property
2456:
2453:
2452:
2451:
2441:
2434:arc-transitive
2422:
2419:chordal graphs
2416:
2410:
2407:perfect graphs
2404:
2401:Petersen graph
2393:
2390:
2347:Main article:
2344:
2341:
2337:disjoint union
2296:Main article:
2293:
2290:
2283:
2274:
2252:
2243:
2232:
2225:
2218:
2205:circular graph
2193:Main article:
2190:
2187:
2175:Main article:
2172:
2169:
2143:
2134:
2123:
2116:
2109:
2084:Main article:
2081:
2078:
2009:Main article:
2006:
2003:
1868:Main article:
1865:
1862:
1855:infinite graph
1842:
1839:
1835:complete graph
1821:Complete graph
1819:Main article:
1816:
1815:Complete graph
1813:
1789:Main article:
1786:
1783:
1748:oriented graph
1743:
1742:Oriented graph
1740:
1738:
1735:
1718:weighted graph
1705:
1704:Weighted graph
1702:
1696:
1689:
1664:
1657:
1604:Main article:
1601:
1598:
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1397:
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1382:
1379:
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1364:
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1199:
1175:
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1128:
1125:
1122:
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1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1061:
1050:directed lines
1046:directed links
1042:directed edges
1027:
978:Multiple edges
959:is called the
888:directed from
867:
866:
854:
851:
848:
845:
831:
827:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
758:directed lines
754:directed links
750:directed edges
735:
694:directed graph
683:directed graph
669:Directed graph
667:Main article:
664:
663:Directed graph
661:
654:
647:
635:
628:
619:
610:
594:
581:, which is an
345:
342:
339:
336:
333:
255:
250:
246:
242:
237:
233:
229:
180:directed graph
159:
156:
147:
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127:owes money to
37:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3891:
3880:
3877:
3876:
3874:
3862:
3861:
3856:
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3848:
3845:
3840:
3836:
3835:
3829:
3826:
3825:
3820:
3815:
3807:
3794:
3788:
3784:
3780:
3779:
3773:
3772:
3768:
3762:
3756:
3752:
3747:
3743:
3737:
3734:. MIT Press.
3732:
3731:
3724:
3720:
3714:
3710:
3705:
3701:
3695:
3691:
3686:
3682:
3676:
3673:. CRC Press.
3672:
3667:
3663:
3657:
3654:. MIT Press.
3653:
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3627:
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3523:
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3502:
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3379:
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3369:
3362:
3359:
3355:
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3338:
3335:
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3298:
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3208:
3204:
3200:
3197: : 284.
3196:
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3159:
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3131:
3130:
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3119:
3113:
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3104:
3101:
3099:
3096:
3094:
3093:Graph drawing
3091:
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3079:
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2866:Cayley graphs
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2571:
2567:
2563:
2560:taking a set
2559:
2555:
2551:
2547:
2543:
2538:
2536:
2532:
2528:
2524:
2520:
2515:
2513:
2509:
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2503:
2499:
2498:trivial graph
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2363:oriented tree
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2359:directed tree
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1798:
1797:regular graph
1792:
1791:Regular graph
1785:Regular graph
1784:
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1189:
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1123:
1120:
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1102:
1099:
1087:
1084:
1081:
1073:
1069:
1062:
1059:
1055:
1051:
1047:
1043:
1040:(also called
1039:
1035:
1028:
1025:
1021:
1018:(also called
1017:
1013:
1006:
1005:
1004:
1000:
996:
992:
988:
982:
980:
979:
972:
968:
962:
961:inverted edge
956:
952:
938:
927:
923:
915:
908:of the edge,
907:
885:
881:
874:
872:
849:
846:
843:
829:
825:
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3566:. Springer.
2512:empty graph
2483:consecutive
2475:consecutive
2333:at most one
2312:exactly one
2264:= 1, 2, …,
2201:cycle graph
2195:Cycle graph
2189:Cycle graph
2155:= 1, 2, …,
2023:partitioned
1851:finite sets
1776:orientation
1612:mixed graph
1606:Mixed graph
1600:Mixed graph
420:empty graph
364:are called
289:of an edge
270:(sometimes
146:Definitions
123:means that
3535:References
3492:2019-09-16
3291:2016-02-16
3269:. p.
3158:vertex set
3003:hypergraph
2932:dual graph
2926:line graph
2651:and edges
2507:null graph
2459:See also:
2378:polyforest
2339:of trees.
2092:path graph
2086:Path graph
2080:Path graph
1928:is called
1886:is called
935:and to be
696:is a pair
602:to vertex
392:or simply
373:multigraph
316:and to be
188:multigraph
3860:MathWorld
3267:CRC Press
3032:structure
3014:simplices
2886:from the
2830:on a set
2531:unlabeled
2319:connected
2207:of order
2098:of order
1888:connected
1500:∈
1482:∣
1461:→
1452:ϕ
1432:ϕ
1399:∈
1381:∣
1360:⊆
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1091:→
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906:endpoints
847:≠
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804:∣
783:⊆
641:symmetric
424:empty set
303:endpoints
3873:Category
3798:8 August
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3486:Archived
3285:Archived
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3184:Archived
3162:edge set
3160:and its
3150:8 August
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3072:See also
2878:, every
2576:Examples
2542:category
2491:incident
2479:adjacent
2471:adjacent
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1584:adjacent
1016:vertices
937:incident
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560:adjacent
366:adjacent
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318:incident
214:vertices
208:, where
107:only if
68:vertices
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381:loops
276:lines
272:links
268:edges
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