2995:
508:
4049:. In these applications, graphs are ordered by specificity, meaning that more constrained graphsâwhich are more specific and thus contain a greater amount of informationâare subsumed by those that are more general. Operations between graphs include evaluating the direction of a subsumption relationship between two graphs, if any, and computing graph unification. The unification of two argument graphs is defined as the most general graph (or the computation thereof) that is consistent with (i.e. contains all of the information in) the inputs, if such a graph exists; efficient unification algorithms are known.
533:
521:
31:
6821:
131:
1524:
3213:
7852:
6833:
3316:, are used to represent structures in which pairwise connections have some numerical values. For example, if a graph represents a road network, the weights could represent the length of each road. There may be several weights associated with each edge, including distance (as in the previous example), travel time, or monetary cost. Such weighted graphs are commonly used to program GPS's, and travel-planning search engines that compare flight times and costs.
7862:
7872:
6857:
6845:
4188:
Decomposition, defined as partitioning the edge set of a graph (with as many vertices as necessary accompanying the edges of each part of the partition), has a wide variety of questions. Often, the problem is to decompose a graph into subgraphs isomorphic to a fixed graph; for instance, decomposing a
3325:
3642:
A graph drawing should not be confused with the graph itself (the abstract, non-visual structure) as there are several ways to structure the graph drawing. All that matters is which vertices are connected to which others by how many edges and not the exact layout. In practice, it is often difficult
3623:
A graph is an abstraction of relationships that emerge in nature; hence, it cannot be coupled to a certain representation. The way it is represented depends on the degree of convenience such representation provides for a certain application. The most common representations are the visual, in which,
3542:
makes fundamental use of the notion of "discharging" developed by Heesch. The proof involved checking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to its complexity. A simpler proof considering only 633 configurations was given twenty years later by
3256:
and conservation efforts where a vertex can represent regions where certain species exist (or inhabit) and the edges represent migration paths or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes
3199:
as a means to model molecules. Graphs and networks are excellent models to study and understand phase transitions and critical phenomena. Removal of nodes or edges leads to a critical transition where the network breaks into small clusters which is studied as a phase transition. This breakdown is
3243:
software. Under the umbrella of social networks are many different types of graphs. Acquaintanceship and friendship graphs describe whether people know each other. Influence graphs model whether certain people can influence the behavior of others. Finally, collaboration graphs model whether two
3638:
Graphs are usually represented visually by drawing a point or circle for every vertex, and drawing a line between two vertices if they are connected by an edge. If the graph is directed, the direction is indicated by drawing an arrow. If the graph is weighted, the weight is added on the arrow.
3997:
Many problems and theorems in graph theory have to do with various ways of coloring graphs. Typically, one is interested in coloring a graph so that no two adjacent vertices have the same color, or with other similar restrictions. One may also consider coloring edges (possibly so that no two
3186:
graphs can be used to represent functional connections between brain areas that interact to give rise to various cognitive processes, where the vertices represent different areas of the brain and the edges represent the connections between those areas. Graph theory plays an important role in
3826:
in a given graph. Again, some important graph properties are hereditary with respect to induced subgraphs, which means that a graph has a property if and only if all induced subgraphs also have it. Finding maximal induced subgraphs of a certain kind is also often NP-complete. For example:
3707:
structures on the other hand provide faster access for some applications but can consume huge amounts of memory. Implementations of sparse matrix structures that are efficient on modern parallel computer architectures are an object of current investigation.
3458:, published in 1969, was "considered the world over to be the definitive textbook on the subject", and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other. Harary donated all of the royalties to fund the
3847:
or subcontraction of a graph is any graph obtained by taking a subgraph and contracting some (or no) edges. Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too. For example,
3276:
and study the relationships between them, such as metabolic pathways and gene regulatory networks. Evolutionary trees, ecological networks, and hierarchical clustering of gene expression patterns are also represented as graph structures.
3002:
Graphs can be used to model many types of relations and processes in physical, biological, social and information systems. Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the term
2411:
1720:
3412:
in 1959. Cayley linked his results on trees with contemporary studies of chemical composition. The fusion of ideas from mathematics with those from chemistry began what has become part of the standard terminology of graph theory.
3187:
electrical modeling of electrical networks, here, weights are associated with resistance of the wire segments to obtain electrical properties of network structures. Graphs are also used to represent the micro-scale channels of
5951:
English edition, Wiley 1961; Methuen & Co, New York 1962; Russian, Moscow 1961; Spanish, Mexico 1962; Roumanian, Bucharest 1969; Chinese, Shanghai 1963; Second printing of the 1962 first
English edition, Dover, New York
3037:
from one page to another. A similar approach can be taken to problems in social media, travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases, and many other fields. The development of
2579:
3698:
used for manipulating the graph. Theoretically one can distinguish between list and matrix structures but in concrete applications the best structure is often a combination of both. List structures are often preferred for
761:
302:
3158:, the three-dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. Also, "the
2783:
2678:
927:
3028:
and non-causal linked structures are graphs that are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a
3469:: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?" This problem was first posed by
3007:
is sometimes defined to mean a graph in which attributes (e.g. names) are associated with the vertices and edges, and the subject that expresses and understands real-world systems as a network is called
5811:
4189:
complete graph into
Hamiltonian cycles. Other problems specify a family of graphs into which a given graph should be decomposed, for instance, a family of cycles, or decomposing a complete graph
5559:
1097:
1009:
5146:
3657:
and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a
3624:
usually, vertices are drawn and connected by edges, and the tabular, in which rows of a table provide information about the relationships between the vertices within the graph.
2217:
853:
596:
3799:
for subgraphs, which means that a graph has the property if and only if all subgraphs have it too. Unfortunately, finding maximal subgraphs of a certain kind is often an
1156:
are usually taken to be finite, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the
3166:
in a form in close contact with the experimental numbers one wants to understand." In chemistry a graph makes a natural model for a molecule, where vertices represent
1582:
1507:
365:
181:
2900:
2698:
2488:
2160:
2124:
1873:
1787:
1437:
1361:
1029:
2313:
1628:
1266:
1232:
1818:
2984:
2964:
2940:
2920:
2868:
2844:
2820:
2599:
2456:
2278:
2240:
2092:
2072:
2048:
2028:
2001:
1977:
1953:
1933:
1913:
1893:
1838:
1605:
1477:
1457:
1405:
1381:
1338:
1198:
1178:
1154:
1134:
947:
809:
657:
619:
496:
476:
452:
432:
405:
385:
204:
3754:, like the adjacency matrix, has both its rows and columns indexed by vertices, but rather than containing a 0 or a 1 in each cell it contains the length of a
3730:, in which both the rows and columns are indexed by vertices. In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The
5338:
5483:
Shah, Preya; Ashourvan, Arian; Mikhail, Fadi; Pines, Adam; Kini, Lohith; Oechsel, Kelly; Das, Sandhitsu R; Stein, Joel M; Shinohara, Russell T (2019-07-01).
5189:
Vecchio, F (2017). ""Small World" architecture in brain connectivity and hippocampal volume in
Alzheimer's disease: a study via graph theory from EEG data".
5047:
Shah, Preya; Ashourvan, Arian; Mikhail, Fadi; Pines, Adam; Kini, Lohith; Oechsel, Kelly; Das, Sandhitsu R; Stein, Joel M; Shinohara, Russell T (2019-07-01).
3182:, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. Similarly, in
3650:
was very influential on the subject of graph drawing. Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings.
3643:
to decide if two drawings represent the same graph. Depending on the problem domain some layouts may be better suited and easier to understand than others.
4601:
3268:
to model and analyse datasets with complex relationships. For example, graph-based methods are often used to 'cluster' cells together into cell-types in
3126:) are common in the analysis of language as a graph. Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as
2998:
The network graph formed by
Knowledge editors (edges) contributing to different Knowledge language versions (vertices) during one month in summer 2013.
2493:
7908:
2438:
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
3819:. It asks whether two graphs are isomorphic. It is not known whether this problem is NP-complete, nor whether it can be solved in polynomial time.
680:
227:
3998:
coincident edges are the same color), or other variations. Among the famous results and conjectures concerning graph coloring are the following:
6281:
5765:
3690:
The tabular representation lends itself well to computational applications. There are different ways to store graphs in a computer system. The
3719:, which separately lists the neighbors of each vertex: Much like the edge list, each vertex has a list of which vertices it is adjacent to.
791:
is an edge that joins a vertex to itself. Graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex
5294:
Kumar, Ankush; Kulkarni, G. U. (2016-01-04). "Evaluating conducting network based transparent electrodes from geometrical considerations".
5642:"Ueber die Analytischen Figuren, welche in der Mathematik BĂ€ume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen"
2703:
2604:
858:
6895:
4012:
4322:
6036:
4317:
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3404:
with particular properties. Enumerative graph theory then arose from the results of Cayley and the fundamental results published by
6225:
3971:
Another class of problems has to do with the extent to which various species and generalizations of graphs are determined by their
3897:
of a graph is any graph obtained by subdividing some (or no) edges. Subdivision containment is related to graph properties such as
4596:
3106:, especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words;
7605:
7577:
3340:
and published in 1736 is regarded as the first paper in the history of graph theory. This paper, as well as the one written by
7901:
7630:
6124:
5974:
5915:
5726:
5397:
5986:
Mathematical results on scale-free random graphs in "Handbook of Graphs and
Networks" (S. Bornholdt and H.G. Schuster (eds))
3578:
came from the use of the techniques of modern algebra. The first example of such a use comes from the work of the physicist
7481:
6644:
3091:
4257:
Many problems involve characterizing the members of various classes of graphs. Some examples of such questions are below:
3359:. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by
7635:
6914:
1034:
952:
6161:
7147:
6789:
6274:
5769:
4033:
3775:: the problem of counting graphs meeting specified conditions. Some of this work is found in Harary and Palmer (1973).
5414:
3341:
3284:; nervous systems can be seen as a graph, where the nodes are neurons and the edges are the connections between them.
7787:
7615:
7152:
6861:
6175:
6030:
6006:
5625:
4970:
4839:
1527:
A directed graph with three vertices and four directed edges (the double arrow represents an edge in each direction).
76:
7951:
7894:
7875:
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6325:
6096:
5347:
4268:
3843:
Still another such problem, the minor containment problem, is to find a fixed graph as a minor of a given graph. A
6185:
7263:
6739:
3364:
7554:
7516:
7180:
6888:
6837:
4814:
4056:, graph unification is the sufficient satisfiability and combination function. Well-known applications include
3938:
3832:
3654:
3544:
3236:
3191:, in which the vertices represent the pores and the edges represent the smaller channels connecting the pores.
3063:
3033:
can be represented by a directed graph, in which the vertices represent web pages and directed edges represent
118:
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related
4089:
3426:, where he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams:
3337:
7696:
7673:
7403:
7393:
6267:
6151:
4053:
3788:
3087:
4179:
The original set cover problem, also called hitting set, can be described as a vertex cover in a hypergraph.
3562:
The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of
7777:
7365:
7273:
7185:
6961:
6946:
6849:
4834:
4819:
4551:
4546:
4288:
4167:
4007:
3954:
3894:
3784:
3556:
3548:
3312:
A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or
3269:
2168:, not allowed under the definition above, are two or more edges with both the same tail and the same head.
52:
7865:
7600:
7105:
6764:
6320:
6146:
5541:
5415:"graphsim: An R package for simulating gene expression data from graph structures of biological pathways"
4682:
4576:
4556:
4354:
4312:
4121:
There are numerous problems arising especially from applications that have to do with various notions of
3661:, the crossing number is zero by definition. Drawings on surfaces other than the plane are also studied.
3583:
5598:
5232:
Vecchio, F (2013). "Brain network connectivity assessed using graph theory in frontotemporal dementia".
811:
to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph)
7837:
7486:
6335:
6052:
4541:
4516:
4109:
3890:
3886:
3685:
3534:
published a method for solving the problem using computers. A computer-aided proof produced in 1976 by
3232:
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6216:
2823:
1341:
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6881:
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3446:
to the product of in- or co-variants whose separate graphs are given. " (italics as in the original).
3409:
3356:
3111:
16:
This article is about sets of vertices connected by edges. For graphs of mathematical functions, see
4521:
2581:. So to allow loops the definitions must be expanded. For directed simple graphs, the definition of
929:. To allow loops, the definitions must be expanded. For undirected simple graphs, the definition of
100:, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in
7706:
7539:
7132:
7001:
6794:
4638:
4307:
4129:
4083:
4073:
4057:
3946:
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3898:
3861:
3726:, a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the
3155:
2178:
814:
557:
113:
6208:
6141:
6114:
5905:
3304:. Algebraic graph theory has been applied to many areas including dynamic systems and complexity.
2458:
to itself is the edge (for a directed simple graph) or is incident on (for a directed multigraph)
7767:
7701:
7592:
7408:
7075:
6679:
6669:
6639:
6573:
6308:
5800:
Heinrich Heesch: Untersuchungen zum
Vierfarbenproblem. Mannheim: Bibliographisches Institut 1969.
5672:
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4566:
4536:
4302:
4235:
3902:
3836:
3379:
3240:
3123:
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57:
4586:
3174:. This approach is especially used in computer processing of molecular structures, ranging from
507:
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7663:
7544:
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7301:
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6654:
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4571:
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of the asymptotic probability of graph connectivity, gave rise to yet another branch, known as
3486:
3478:
3417:
3297:
3099:
3067:
1276:
of a vertex is the number of edges that are incident to it, where a loop is counted twice. The
119:
21:
8005:
7802:
7772:
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7658:
7572:
7448:
7388:
7355:
7345:
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7127:
6996:
6971:
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6931:
6804:
6734:
6611:
6535:
6474:
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6454:
6431:
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4104:
4094:
4078:
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3523:
3401:
3387:
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3086:
and compositional semantics follow tree-based structures, whose expressive power lies in the
1549:
1486:
338:
148:
62:
5277:
4789:
3459:
2406:{\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}
1715:{\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}
7971:
7562:
7534:
7506:
7501:
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7243:
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6583:
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Phase
Transitions in Combinatorial Optimization Problems, Section 3: Introduction to Graphs
6017:
5684:
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5013:
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4729:
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4359:
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4141:
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2683:
2461:
2133:
2097:
1846:
1765:
1410:
1346:
1014:
101:
5846:
Appel, K.; Haken, W. (1977), "Every planar map is four colorable. Part II. Reducibility",
8:
7827:
7752:
7668:
7653:
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7205:
7162:
7157:
7054:
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6709:
6631:
6530:
6464:
6421:
6411:
6391:
4606:
4591:
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4439:
4273:
Ascertaining relationships among classes (e.g. does one property of graphs imply another)
4241:
3849:
3800:
3502:
3392:
3179:
2434:
1241:
1207:
787:
17:
6222:
5688:
5433:
5307:
5017:
4954:
3664:
There are other techniques to visualize a graph away from vertices and edges, including
1800:
60:
used to model pairwise relations between objects. A graph in this context is made up of
7792:
7691:
7567:
7524:
7433:
7375:
7360:
7350:
7142:
6941:
6825:
6744:
6684:
6616:
6606:
6545:
6520:
6396:
6353:
6348:
5995:
5517:
5484:
5465:
5257:
5214:
5171:
5081:
5048:
5029:
5003:
4976:
4940:
4774:
4754:
4734:
4704:
4531:
4459:
4424:
4379:
4225:
4027:
4002:
3669:
3490:
3474:
3466:
3273:
3201:
3082:, since natural language often lends itself well to discrete structure. Traditionally,
2969:
2949:
2925:
2905:
2853:
2829:
2805:
2584:
2441:
2263:
2225:
2077:
2057:
2033:
2013:
1986:
1962:
1938:
1918:
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1823:
1590:
1462:
1442:
1390:
1366:
1323:
1183:
1163:
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1119:
932:
794:
642:
604:
532:
520:
481:
461:
437:
417:
390:
370:
189:
30:
5875:
Robertson, N.; Sanders, D.; Seymour, P.; Thomas, R. (1997), "The four color theorem",
4829:
3885:
A similar problem, the subdivision containment problem, is to find a fixed graph as a
3601:
The introduction of probabilistic methods in graph theory, especially in the study of
3438:
diagram or chemicograph. I give a rule for the geometrical multiplication of graphs,
7812:
7742:
7721:
7683:
7491:
7458:
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Fractal Music, Hypercards, and moreâŠMathematical
Recreations from Scientific American
5722:
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led to the study of the colorings of the graphs embedded on surfaces with arbitrary
3481:
the same year. Many incorrect proofs have been proposed, including those by Cayley,
3292:
In mathematics, graphs are useful in geometry and certain parts of topology such as
7979:
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7645:
7529:
7496:
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is a modified form of the adjacency matrix that incorporates information about the
3735:
3727:
3723:
3673:
3595:
3591:
3579:
3514:
3451:
3422:
3346:
3107:
3021:
1523:
6190:
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3435:
498:. A vertex may exist in a graph and not belong to an edge. Under this definition,
7716:
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7582:
7476:
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7398:
7253:
7248:
7195:
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7011:
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3470:
3217:
3196:
3175:
3159:
3047:
3009:
2574:{\displaystyle \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}
6196:
5147:"A social network analysis of Twitter: Mapping the digital humanities community"
4994:
Mashaghi, A.; et al. (2004). "Investigation of a protein complex network".
3517:. The works of Ramsey on colorations and more specially the results obtained by
3405:
7822:
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6772:
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2164:
1518:
1157:
768:
500:
321:
5764:
5442:
5202:
4228:, a decomposition into a collection of cycles covering each edge exactly twice
4045:
Constraint modeling theories concern families of directed graphs related by a
3791:
in a given graph. One reason to be interested in such a question is that many
3244:
people work together in a particular way, such as acting in a movie together.
3078:
Graph-theoretic methods, in various forms, have proven particularly useful in
7999:
7923:
7711:
7006:
6560:
6492:
6444:
6254:
6158:
5874:
5860:
5831:
5657:
5593:
5508:
5461:
5323:
5131:
5072:
4824:
4764:
4749:
4665:
4650:
4469:
4464:
4434:
4399:
4374:
4245:
4231:
4176:
is the special case of set cover problem where sets to cover are every edges.
4046:
3755:
3747:
3731:
3633:
3535:
3498:
3383:
3212:
3188:
3171:
3119:
756:{\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}
297:{\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}
35:
5778:
4962:
4739:
3602:
3375:
7807:
7466:
6502:
6401:
5889:
5526:
5500:
5253:
5210:
5090:
5064:
4779:
4769:
4699:
4655:
4633:
4166:
problem is the special case of set cover problem where sets are the closed
4122:
4099:
4021:
3950:
3910:
3857:
3700:
3658:
3611:
3482:
3455:
3301:
3281:
2418:
1751:
143:
7886:
6259:
5485:"Characterizing the role of the structural connectome in seizure dynamics"
5372:"Social network analysis and visualization: Morenoâs Sociograms revisited"
5049:"Characterizing the role of the structural connectome in seizure dynamics"
3530:
The four color problem remained unsolved for more than a century. In 1969
7984:
7797:
7423:
7335:
6704:
6368:
6291:
6242:
5712:
5008:
4849:
4660:
3647:
3293:
3079:
3042:
to handle graphs is therefore of major interest in computer science. The
3025:
1110:
504:, in which two or more edges connect the same vertices, are not allowed.
43:
6248:
5452:
3653:
Graph drawing also can be said to encompass problems that deal with the
3090:, modeled in a hierarchical graph. More contemporary approaches such as
2421:
of vertices (that is, an edge is associated with two distinct vertices).
1754:
of vertices (that is, an edge is associated with two distinct vertices).
771:
of vertices (that is, an edge is associated with two distinct vertices).
511:
Example of simple undirected graph with 3 vertices, 3 edges and 4 loops.
324:
of vertices (that is, an edge is associated with two distinct vertices).
7817:
7747:
7340:
7080:
6936:
6689:
6568:
6363:
5557:
Cauchy, A. L. (1813), "Recherche sur les polyÚdres - premier mémoire",
4677:
4449:
4219:
3465:
One of the most famous and stimulating problems in graph theory is the
2785:. To avoid ambiguity, these types of objects may be called precisely a
778:
5315:
7931:
7322:
7283:
5697:
4724:
4694:
4277:
4215:
Some specific decomposition problems that have been studied include:
3712:
3695:
3574:. Another important factor of common development of graph theory and
3397:
3367:, and represents the beginning of the branch of mathematics known as
3324:
3228:
3220:
3147:
3039:
3034:
5641:
3054:
systems focusing on rule-based in-memory manipulation of graphs are
2094:. A vertex may exist in a graph and not belong to an edge. The edge
7956:
7936:
7383:
6873:
6593:
6512:
6439:
5574:
L'Huillier, S.-A.-J. (1812â1813), "MĂ©moire sur la polyĂšdromĂ©trie",
3575:
3386:
was led by an interest in particular analytical forms arising from
3368:
3265:
775:
To avoid ambiguity, this type of object may be called precisely an
538:
For vertices U,V,W and X, the degrees are 2,2,3 and 1 respectively.
328:
To avoid ambiguity, this type of object may be called precisely an
6202:
6019:
Graph Theory with
Applications to Engineering and Computer Science
4945:
3485:, and others. The study and the generalization of this problem by
2778:{\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\right\}}
2673:{\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\right\}}
2425:
To avoid ambiguity, this type of object may be called precisely a
1758:
To avoid ambiguity, this type of object may be called precisely a
922:{\displaystyle \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}
6378:
6186:
Concise, annotated list of graph theory resources for researchers
4918:
4916:
4061:
3587:
3253:
3151:
3135:
3131:
3030:
2171:
In one more general sense of the term allowing multiple edges, a
550:
In one more general sense of the term allowing multiple edges, a
5937:
Lists, Decisions and Graphs. With an
Introduction to Probability
7946:
6199:â non-technical paper discussing graphs of people and computers
3615:, which has been a fruitful source of graph-theoretic results.
3430:" Every invariant and co-variant thus becomes expressible by a
3083:
4935:
Hale, Scott A. (2014). "Multilinguals and Knowledge editing".
4913:
4892:
4867:
3505:. Tait's reformulation generated a new class of problems, the
526:
For vertices A,B,C and D, the degrees are respectively 4,4,5,1
7941:
5573:
3521:
in 1941 was at the origin of another branch of graph theory,
3127:
1794:
5539:
3742:
of the vertices, and is useful in some calculations such as
3374:
More than one century after Euler's paper on the bridges of
5107:"Applications of Graph Theory [Scanning the Issue]"
3778:
3167:
1099:. To avoid ambiguity, these types of objects may be called
6232:
with references and links to graph library implementations
1280:
of a graph is the maximum of the degrees of its vertices.
5812:"Every planar map is four colorable. Part I. Discharging"
5482:
5046:
1320:
The edges of an undirected simple graph permitting loops
3473:
in 1852 and its first written record is in a letter of
1542:
In one restricted but very common sense of the term, a
138:
In one restricted but very common sense of the term, a
5374:. Redesigned network strictly based on Moreno (1934),
3122:) and morphology (e.g. finite-state morphology, using
2802:
The edges of a directed simple graph permitting loops
4937:
Proceedings of the 2014 ACM conference on Web science
2972:
2952:
2928:
2908:
2876:
2856:
2832:
2808:
2706:
2686:
2607:
2587:
2496:
2464:
2444:
2316:
2266:
2228:
2181:
2136:
2100:
2080:
2060:
2036:
2016:
1989:
1965:
1941:
1921:
1901:
1881:
1849:
1826:
1803:
1768:
1631:
1593:
1552:
1489:
1465:
1445:
1413:
1393:
1369:
1349:
1326:
1244:
1210:
1186:
1166:
1142:
1122:
1037:
1017:
955:
935:
861:
817:
797:
683:
645:
607:
560:
484:
464:
440:
420:
393:
373:
341:
230:
192:
151:
5599:"On the theory of the analytical forms called trees"
4205:
specified trees having, respectively, 1, 2, 3, ...,
3807:
Finding the largest complete subgraph is called the
3396:. This study had many implications for theoretical
1092:{\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\}}
1004:{\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\}}
96:, where edges link two vertices symmetrically, and
6116:Graph Algorithms in The Language of Linear Algebra
5994:
5907:Graph Algorithms in the Language of Linear Algebra
5845:
5809:
5597:
5276:
5105:
3937:Another problem in subdivision containment is the
3450:The first textbook on graph theory was written by
3416:In particular, the term "graph" was introduced by
2978:
2958:
2934:
2914:
2894:
2862:
2838:
2814:
2777:
2692:
2672:
2593:
2573:
2482:
2450:
2405:
2272:
2234:
2211:
2154:
2118:
2086:
2066:
2042:
2022:
1995:
1971:
1947:
1927:
1907:
1887:
1867:
1832:
1812:
1781:
1714:
1599:
1576:
1501:
1471:
1451:
1431:
1399:
1375:
1355:
1332:
1260:
1226:
1192:
1172:
1148:
1128:
1091:
1023:
1003:
941:
921:
847:
803:
755:
651:
613:
590:
490:
470:
446:
426:
399:
379:
359:
296:
198:
175:
4222:, a decomposition into as few forests as possible
3831:Finding the largest edgeless induced subgraph or
3694:used depends on both the graph structure and the
3679:
3408:between 1935 and 1937. These were generalized by
3272:. Another use is to model genes or proteins in a
7997:
5955:
5933:
4922:
4898:
4873:
4602:Tarjan's strongly connected components algorithm
3815:One special case of subgraph isomorphism is the
3146:Graph theory is also used to study molecules in
1011:. For undirected multigraphs, the definition of
6171:A searchable database of small connected graphs
5983:
5934:Bender, Edward A.; Williamson, S. Gill (2010).
4644:
5766:Society for Industrial and Applied Mathematics
5646:Berichte der Deutschen Chemischen Gesellschaft
4040:
2680:. For directed multigraphs, the definition of
7902:
6913:Note: This template roughly follows the 2012
6889:
6275:
6112:
5903:
5556:
5293:
4052:For constraint frameworks which are strictly
3130:, as well as various 'Net' projects, such as
1539:is a graph in which edges have orientations.
6251:2007 by Jorgen Bang-Jensen and Gregory Gutin
6249:Digraphs: Theory Algorithms and Applications
6090:
6081:
5274:
4688:
3913:if it contains as a subdivision neither the
3400:. The techniques he used mainly concern the
1840:elements that are not necessarily distinct.
1086:
1065:
1053:
1050:
998:
977:
965:
962:
916:
877:
865:
862:
842:
836:
830:
818:
750:
711:
699:
696:
354:
342:
291:
252:
240:
237:
134:A graph with three vertices and three edges.
7916:
6289:
6062:Algorithmic Graph Theory and Perfect Graphs
5771:Looking Back, Looking Ahead: A SIAM History
5103:
4086:(also called the "Chinese postman problem")
3390:to study a particular class of graphs, the
3094:model the syntax of natural language using
544:Examples of finding the degree of vertices.
7909:
7895:
6896:
6882:
6282:
6268:
5964:
5721:, Cambridge University Press, p. 30,
5639:
5412:
5104:Adali, Tulay; Ortega, Antonio (May 2018).
3703:as they have smaller memory requirements.
3257:to the movement can affect other species.
3114:. Still, other methods in phonology (e.g.
2556:
2548:
2388:
2380:
1697:
1689:
906:
898:
740:
732:
281:
273:
6091:Mahadev, N. V. R.; Peled, Uri N. (1995).
6077:. Reading, Massachusetts: Addison-Wesley.
5992:
5956:Biggs, N.; Lloyd, E.; Wilson, R. (1986).
5888:
5877:Journal of Combinatorial Theory, Series B
5859:
5830:
5696:
5670:
5664:
5567:
5540:Biggs, N.; Lloyd, E.; Wilson, R. (1986),
5516:
5451:
5441:
5390:Discrete mathematics and its applications
5346:. Oxford University Press. Archived from
5165:
5144:
5080:
5007:
4944:
4318:List of unsolved problems in graph theory
4183:
3715:, an array of pairs of vertices, and the
3454:, and published in 1936. Another book by
3382:was introducing the concept of topology,
6082:Harary, Frank; Palmer, Edgar M. (1973).
6059:
6025:. Englewood, New Jersey: Prentice-Hall.
5754:, W. H. Freeman and Company, p. 203
5413:Kelly, S.; Black, Michael (2020-07-09).
4993:
3779:Subgraphs, induced subgraphs, and minors
3627:
3323:
3211:
3141:
2993:
1522:
1101:undirected simple graph permitting loops
506:
129:
29:
6205:â tools to teach and learn graph theory
6046:
5984:BollobĂĄs, BĂ©la; Riordan, O. M. (2003).
5947:Théorie des graphes et ses applications
5746:
5633:
5392:(7th ed.). New York: McGraw-Hill.
5231:
5188:
4910:See, for instance, Graham et al., p. 5.
4885:See, for instance, Iyanaga and Kawada,
4248:into regular subgraphs of given degrees
3160:Feynman graphs and rules of calculation
3046:is often formalized and represented by
1287:, the maximum degree of each vertex is
1283:In an undirected simple graph of order
38:of a graph with 6 vertices and 7 edges.
7998:
7606:Knowledge representation and reasoning
6113:Kepner, Jeremy; Gilbert, John (2011).
6103:
6072:
5965:Bondy, J. A.; Murty, U. S. R. (2008).
5904:Kepner, Jeremy; Gilbert, John (2011).
5592:
5550:
5336:
5283:. New York: McGraw-Hill. p. viii.
4135:
3860:if it contains as a minor neither the
3734:indicates the degree of vertices. The
2787:directed simple graph permitting loops
1180:is often assumed to be non-empty, but
1105:undirected multigraph permitting loops
7890:
7631:Philosophy of artificial intelligence
6877:
6263:
6086:. New York, New York: Academic Press.
5944:
5711:
5533:
5387:
5275:Bjorken, J. D.; Drell, S. D. (1965).
1294:and the maximum size of the graph is
6957:Energy consumption (Green computing)
6903:
6844:
6119:. Philadelphia, Pennsylvania: SIAM.
5988:(1st ed.). Weinheim: Wiley VCH.
5868:
4934:
4147:
3252:Likewise, graph theory is useful in
3227:Graph theory is also widely used in
3092:head-driven phrase structure grammar
2791:directed multigraph permitting loops
1200:is allowed to be the empty set. The
7636:Distributed artificial intelligence
6915:ACM Computing Classification System
6856:
6236:
6093:Threshold Graphs and Related Topics
6015:
5839:
5803:
4597:Pushârelabel maximum flow algorithm
4267:Characterizing a class in terms of
4062:elaboration of linguistic structure
3015:
13:
7148:Integrated development environment
5768:(2002), "The George Polya Prize",
4671:
4159:on subsets of vertices/subgraphs.
4034:Hadwiger conjecture (graph theory)
3672:, and other visualizations of the
3270:single-cell transcriptome analysis
3207:
1762:. In set theory and graph theory,
14:
8017:
7616:Automated planning and scheduling
7153:Software configuration management
6255:Graph Theory, by Reinhard Diestel
6134:
4328:
4067:
3986:
3618:
3260:Graphs are also commonly used in
2946:to one another, which is denoted
2007:of the edge. The edge is said to
1512:
1483:to one another, which is denoted
411:of the edge. The edge is said to
92:). A distinction is made between
7952:Microsoft Automatic Graph Layout
7870:
7860:
7851:
7850:
6855:
6843:
6832:
6831:
6819:
6182: (archived February 6, 2006)
6042:from the original on 2019-05-17.
5671:Sylvester, James Joseph (1878).
5560:Journal de l'Ăcole Polytechnique
5388:Rosen, Kenneth H. (2011-06-14).
4252:
3787:, is finding a fixed graph as a
3420:in a paper published in 1878 in
531:
519:
7861:
7264:Computational complexity theory
6740:Computational complexity theory
5897:
5794:
5758:
5740:
5705:
5586:
5476:
5422:Journal of Open Source Software
5406:
5381:
5364:
5330:
5287:
5268:
5225:
5182:
5138:
4155:in graphs may refer to various
4116:
3771:There is a large literature on
3307:
2989:
7055:Network performance evaluation
5428:(51). The Open Journal: 2161.
5097:
5040:
4987:
4928:
4904:
4879:
4234:, a decomposition into as few
3766:
3722:Matrix structures include the
3680:Tabular: Graph data structures
3287:
3073:
2889:
2877:
2870:. Specifically, for each edge
2754:
2742:
2736:
2724:
2716:
2649:
2637:
2631:
2619:
2532:
2520:
2514:
2502:
2477:
2465:
2364:
2352:
2346:
2334:
2326:
2206:
2188:
2149:
2137:
2113:
2101:
1862:
1850:
1820:that is, ordered sequences of
1673:
1661:
1655:
1643:
1571:
1559:
1426:
1414:
1407:. Specifically, for each edge
1254:
1246:
1234:, its number of vertices. The
1220:
1212:
1047:
693:
585:
567:
170:
158:
107:
1:
7419:Multimedia information system
7404:Geographic information system
7394:Enterprise information system
6990:Computer systems organization
5927:
5810:Appel, K.; Haken, W. (1977),
5167:10.1080/23311983.2016.1171458
4510:
4013:ErdĆsâFaberâLovĂĄsz conjecture
3822:A similar problem is finding
3783:A common problem, called the
3328:The Königsberg Bridge problem
3280:Graph theory is also used in
3239:, notably through the use of
3088:principle of compositionality
2212:{\displaystyle G=(V,E,\phi )}
848:{\displaystyle \{x,x\}=\{x\}}
591:{\displaystyle G=(V,E,\phi )}
7778:Computational social science
7366:Theoretical computer science
7186:Software development process
6962:Electronic design automation
6947:Very Large Scale Integration
6245:(2006) by Hartmann and Weigt
5777:, p. 26, archived from
5618:10.1017/CBO9780511703690.046
5246:10.1212/WNL.0b013e31829a33f8
5154:Cogent Arts & Humanities
4923:Bender & Williamson 2010
4899:Bender & Williamson 2010
4874:Bender & Williamson 2010
4645:Related areas of mathematics
4587:Planarity testing algorithms
4323:Publications in graph theory
4008:Strong perfect graph theorem
3785:subgraph isomorphism problem
3711:List structures include the
3582:, who published in 1845 his
27:Area of discrete mathematics
7:
7601:Natural language processing
7389:Information storage systems
6147:Encyclopedia of Mathematics
5279:Relativistic Quantum Fields
4996:European Physical Journal B
4683:Abstract simplicial complex
4612:
4577:Nearest neighbour algorithm
4355:Disjoint-set data structure
4313:List of graph theory topics
4295:
4090:Seven bridges of Königsberg
4041:Subsumption and unification
3761:
3434:precisely identical with a
3338:Seven Bridges of Königsberg
3216:Graph theory in sociology:
3110:are therefore important in
1268:, its number of edges. The
10:
8022:
7517:Humanâcomputer interaction
7487:Intrusion detection system
7399:Social information systems
7384:Database management system
6790:Films about mathematicians
6223:A list of graph algorithms
6197:The Social Life of Routers
6108:. Oxford University Press.
6053:Cambridge University Press
5960:. Oxford University Press.
5370:Grandjean, Martin (2015).
5296:Journal of Applied Physics
5191:Brain Imaging and Behavior
5145:Grandjean, Martin (2016).
5124:10.1109/JPROC.2018.2820300
5026:10.1140/epjb/e2004-00301-0
4634:Probabilistic graph theory
4110:Traveling salesman problem
3990:
3939:KelmansâSeymour conjecture
3686:Graph (abstract data type)
3683:
3631:
3509:, particularly studied by
3319:
3247:
3231:as a way, for example, to
3184:computational neuroscience
3178:to database searching. In
1516:
111:
15:
7970:
7922:
7846:
7783:Computational engineering
7758:Computational mathematics
7735:
7682:
7644:
7591:
7553:
7515:
7457:
7374:
7320:
7282:
7234:
7171:
7104:
7068:
7025:
6989:
6922:
6911:
6813:
6763:
6720:
6630:
6592:
6559:
6511:
6483:
6430:
6377:
6359:Philosophy of mathematics
6334:
6299:
6106:Networks: An Introduction
6060:Golumbic, Martin (1980).
5997:Introductory Graph Theory
5910:. SIAM. p. 1171458.
5546:, Oxford University Press
5443:10.1101/2020.03.02.972471
5340:Networks: An Introduction
5203:10.1007/s11682-016-9528-3
4689:Prominent graph theorists
4582:Network simplex algorithm
4058:automatic theorem proving
4018:Total coloring conjecture
3981:reconstruction conjecture
3874:) nor the complete graph
3817:graph isomorphism problem
3112:computational linguistics
2417:mapping every edge to an
767:mapping every edge to an
74:) which are connected by
7793:Computational healthcare
7788:Differentiable computing
7707:Graphics processing unit
7133:Domain-specific language
7002:Computational complexity
6795:Recreational mathematics
6211:, and library resources
6049:Algorithmic Graph Theory
5993:Chartrand, Gary (1985).
5658:10.1002/cber.18750080252
5576:Annales de Mathématiques
4889:, p. 234 or Biggs, p. 4.
4861:
4639:Topological graph theory
4552:FordâFulkerson algorithm
4547:FloydâWarshall algorithm
4308:Glossary of graph theory
4130:Max flow min cut theorem
4084:Route inspection problem
4074:Hamiltonian path problem
4028:List coloring conjecture
4024:'s conjecture (unsolved)
3915:complete bipartite graph
3862:complete bipartite graph
3584:Kirchhoff's circuit laws
3233:measure actors' prestige
3156:condensed matter physics
3124:finite-state transducers
3096:typed feature structures
3066:storing and querying of
3044:transformation of graphs
125:
114:Glossary of graph theory
7917:Graph analysis software
7768:Computational chemistry
7702:Photograph manipulation
7593:Artificial intelligence
7409:Decision support system
6680:Mathematical statistics
6670:Mathematical psychology
6640:Engineering mathematics
6574:Algebraic number theory
5958:Graph Theory, 1736â1936
5673:"Chemistry and Algebra"
5563:, 9 (Cahier 16): 66â86.
5543:Graph Theory, 1736-1936
5112:Proceedings of the IEEE
4963:10.1145/2615569.2615684
4557:HopcroftâKarp algorithm
4490:Strongly regular graphs
4303:Gallery of named graphs
4269:forbidden substructures
4244:, a decomposition of a
3973:point-deleted subgraphs
3837:independent set problem
3646:The pioneering work of
3241:social network analysis
3100:directed acyclic graphs
1577:{\displaystyle G=(V,E)}
1502:{\displaystyle x\sim y}
360:{\displaystyle \{x,y\}}
330:undirected simple graph
176:{\displaystyle G=(V,E)}
120:mathematical structures
58:mathematical structures
7833:Educational technology
7664:Reinforcement learning
7414:Process control system
7312:Computational geometry
7302:Algorithmic efficiency
7297:Analysis of algorithms
6952:Systems on Chip (SoCs)
6826:Mathematics portal
6675:Mathematical sociology
6655:Mathematical economics
6650:Mathematical chemistry
6579:Analytic number theory
6460:Differential equations
6073:Harary, Frank (1969).
6047:Gibbons, Alan (1985).
6016:Deo, Narsingh (1974).
5945:Berge, Claude (1958).
5890:10.1006/jctb.1997.1750
5861:10.1215/ijm/1256049012
5832:10.1215/ijm/1256049011
5605:Philosophical Magazine
4624:Geometric graph theory
4619:Algebraic graph theory
4542:EdmondsâKarp algorithm
4517:BellmanâFord algorithm
4455:Pebble motion problems
4415:Graph sandwich problem
4335:Algebraic graph theory
4291:for members of a class
4264:the members of a class
4184:Decomposition problems
3758:between two vertices.
3507:factorization problems
3329:
3298:Algebraic graph theory
3224:
2999:
2980:
2960:
2936:
2916:
2896:
2864:
2840:
2816:
2779:
2700:should be modified to
2694:
2674:
2601:should be modified to
2595:
2575:
2484:
2452:
2407:
2274:
2236:
2213:
2156:
2120:
2088:
2068:
2044:
2024:
1997:
1973:
1949:
1929:
1909:
1889:
1869:
1834:
1814:
1783:
1716:
1601:
1578:
1528:
1503:
1473:
1453:
1433:
1401:
1377:
1357:
1334:
1262:
1228:
1194:
1174:
1150:
1130:
1093:
1031:should be modified to
1025:
1005:
949:should be modified to
943:
923:
849:
805:
757:
653:
615:
592:
512:
492:
472:
448:
428:
401:
381:
361:
298:
200:
177:
135:
39:
22:Graph (disambiguation)
20:. For other uses, see
7803:Electronic publishing
7773:Computational biology
7763:Computational physics
7659:Unsupervised learning
7573:Distributed computing
7449:Information retrieval
7356:Mathematical analysis
7346:Mathematical software
7236:Theory of computation
7201:Software construction
7191:Requirements analysis
7069:Software organization
6997:Computer architecture
6967:Hardware acceleration
6932:Printed circuit board
6805:Mathematics education
6735:Theory of computation
6455:Hypercomplex analysis
6203:Graph Theory Software
6176:Image gallery: graphs
6159:Graph theory tutorial
6104:Newman, Mark (2010).
6084:Graphical Enumeration
5337:Newman, Mark (2010).
4730:Dirac, Gabriel Andrew
4629:Extremal graph theory
4485:Spectral graph theory
4475:Random regular graphs
4284:membership in a class
4105:Three-cottage problem
4095:Shortest path problem
4079:Minimum spanning tree
3872:Three-cottage problem
3773:graphical enumeration
3628:Visual: Graph drawing
3524:extremal graph theory
3402:enumeration of graphs
3388:differential calculus
3332:The paper written by
3327:
3300:has close links with
3215:
3193:Chemical graph theory
3142:Physics and chemistry
3068:graph-structured data
3048:graph rewrite systems
2997:
2981:
2961:
2937:
2917:
2897:
2895:{\displaystyle (x,y)}
2865:
2841:
2826:~ on the vertices of
2817:
2780:
2695:
2693:{\displaystyle \phi }
2675:
2596:
2576:
2485:
2483:{\displaystyle (x,x)}
2453:
2408:
2275:
2237:
2214:
2175:is an ordered triple
2157:
2155:{\displaystyle (x,y)}
2121:
2119:{\displaystyle (y,x)}
2089:
2069:
2045:
2025:
1998:
1974:
1950:
1930:
1910:
1890:
1870:
1868:{\displaystyle (x,y)}
1835:
1815:
1784:
1782:{\displaystyle V^{n}}
1760:directed simple graph
1717:
1602:
1579:
1526:
1504:
1474:
1454:
1434:
1432:{\displaystyle (x,y)}
1402:
1378:
1358:
1356:{\displaystyle \sim }
1335:
1263:
1229:
1195:
1175:
1151:
1131:
1094:
1026:
1024:{\displaystyle \phi }
1006:
944:
924:
850:
806:
758:
654:
616:
593:
554:is an ordered triple
510:
493:
473:
449:
429:
402:
382:
362:
299:
201:
178:
133:
112:Further information:
33:
7563:Concurrent computing
7535:Ubiquitous computing
7507:Application security
7502:Information security
7331:Discrete mathematics
7307:Randomized algorithm
7259:Computability theory
7244:Model of computation
7216:Software maintenance
7211:Software engineering
7173:Software development
7123:Programming language
7118:Programming paradigm
7035:Network architecture
6785:Informal mathematics
6665:Mathematical physics
6660:Mathematical finance
6645:Mathematical biology
6584:Diophantine geometry
6193:â a graph theory IDE
5501:10.1093/brain/awz125
5065:10.1093/brain/awz125
4567:Kosaraju's algorithm
4537:Dijkstra's algorithm
4527:Breadth-first search
4500:Transitive reduction
4395:Graph data structure
4360:Dual-phase evolution
4174:Vertex cover problem
4142:Museum guard problem
3903:Kuratowski's Theorem
3889:of a given graph. A
3586:for calculating the
3351:carried on with the
3164:quantum field theory
3052:graph transformation
2970:
2950:
2926:
2906:
2874:
2854:
2830:
2824:homogeneous relation
2806:
2704:
2684:
2605:
2585:
2494:
2462:
2442:
2314:
2264:
2226:
2179:
2134:
2098:
2078:
2058:
2034:
2014:
1987:
1963:
1939:
1919:
1899:
1879:
1847:
1824:
1801:
1766:
1629:
1591:
1550:
1487:
1463:
1443:
1411:
1391:
1367:
1347:
1342:homogeneous relation
1324:
1242:
1208:
1184:
1164:
1140:
1120:
1035:
1015:
953:
933:
859:
815:
795:
681:
643:
605:
558:
482:
462:
438:
418:
391:
371:
339:
228:
190:
149:
102:discrete mathematics
7838:Document management
7828:Operations research
7753:Enterprise software
7669:Multi-task learning
7654:Supervised learning
7376:Information systems
7206:Software deployment
7163:Software repository
7017:Real-time computing
6800:Mathematics and art
6710:Operations research
6465:Functional analysis
5689:1878Natur..17..284S
5640:Cayley, A. (1875),
5453:10.21105/joss.02161
5434:2020JOSS....5.2161K
5308:2016JAP...119a5102K
5018:2004EPJB...41..113M
4955:2013arXiv1312.0976H
4939:. pp. 99â108.
4775:Heawood, Percy John
4755:Fleischner, Herbert
4735:Dijkstra, Edsger W.
4607:Topological sorting
4572:Kruskal's algorithm
4562:Hungarian algorithm
4522:BorĆŻvka's algorithm
4505:Tree data structure
4242:Graph factorization
4136:Visibility problems
3801:NP-complete problem
3744:Kirchhoff's theorem
3612:random graph theory
3442:for constructing a
3180:statistical physics
3050:. Complementary to
2846:that is called the
2427:directed multigraph
1789:denotes the set of
1546:is an ordered pair
1383:that is called the
1363:on the vertices of
1340:induce a symmetric
1261:{\displaystyle |E|}
1227:{\displaystyle |V|}
18:Graph of a function
7621:Search methodology
7568:Parallel computing
7525:Interaction design
7434:Computing platform
7361:Numerical analysis
7351:Information theory
7143:Software framework
7106:Software notations
7045:Network components
6942:Integrated circuit
6745:Numerical analysis
6354:Mathematical logic
6349:Information theory
6228:2019-07-13 at the
6219:about graph theory
6217:in other libraries
6164:2012-01-16 at the
4840:Thomassen, Carsten
4800:NeĆĄetĆil, Jaroslav
4715:Brightwell, Graham
4710:Bondy, Adrian John
4532:Depth-first search
4460:Percolation theory
4425:Intersection graph
4380:Graph automorphism
4276:Finding efficient
4226:Cycle double cover
4157:set cover problems
4003:Four-color theorem
3949:graph that is not
3947:5-vertex-connected
3670:intersection graph
3467:four color problem
3330:
3225:
3202:percolation theory
3000:
2976:
2956:
2932:
2912:
2892:
2860:
2848:adjacency relation
2836:
2812:
2775:
2690:
2670:
2591:
2571:
2480:
2448:
2415:incidence function
2403:
2270:
2232:
2209:
2152:
2116:
2084:
2064:
2040:
2020:
1993:
1969:
1945:
1925:
1905:
1885:
1865:
1830:
1813:{\displaystyle V,}
1810:
1779:
1712:
1597:
1574:
1529:
1499:
1469:
1449:
1429:
1397:
1385:adjacency relation
1373:
1353:
1330:
1258:
1224:
1190:
1170:
1146:
1126:
1089:
1021:
1001:
939:
919:
845:
801:
765:incidence function
753:
649:
611:
588:
513:
488:
468:
444:
424:
397:
377:
357:
294:
196:
173:
136:
40:
7993:
7992:
7884:
7883:
7813:Electronic voting
7743:Quantum Computing
7736:Applied computing
7722:Image compression
7492:Hardware security
7482:Security services
7439:Digital marketing
7226:Open-source model
7138:Modeling language
7050:Network scheduler
6871:
6870:
6470:Harmonic analysis
6126:978-0-898719-90-1
5976:978-1-84628-969-9
5917:978-0-898719-90-1
5848:Illinois J. Math.
5819:Illinois J. Math.
5728:978-0-521-79489-3
5399:978-0-07-338309-5
5376:Who Shall Survive
5316:10.1063/1.4939280
4720:Chudnovsky, Maria
4480:Semantic networks
4370:Existential graph
4153:Covering problems
4148:Covering problems
4123:flows in networks
4060:and modeling the
3824:induced subgraphs
3746:on the number of
3596:electric circuits
3262:molecular biology
3116:optimality theory
3108:semantic networks
3104:lexical semantics
2979:{\displaystyle y}
2959:{\displaystyle x}
2935:{\displaystyle y}
2915:{\displaystyle x}
2863:{\displaystyle G}
2839:{\displaystyle G}
2815:{\displaystyle G}
2594:{\displaystyle E}
2553:
2451:{\displaystyle x}
2385:
2273:{\displaystyle E}
2235:{\displaystyle V}
2087:{\displaystyle y}
2067:{\displaystyle x}
2043:{\displaystyle y}
2023:{\displaystyle x}
1996:{\displaystyle y}
1972:{\displaystyle x}
1948:{\displaystyle y}
1928:{\displaystyle x}
1908:{\displaystyle y}
1888:{\displaystyle x}
1833:{\displaystyle n}
1694:
1600:{\displaystyle V}
1472:{\displaystyle y}
1452:{\displaystyle x}
1400:{\displaystyle G}
1376:{\displaystyle G}
1333:{\displaystyle G}
1193:{\displaystyle E}
1173:{\displaystyle V}
1149:{\displaystyle E}
1129:{\displaystyle V}
1114:), respectively.
942:{\displaystyle E}
903:
804:{\displaystyle x}
737:
652:{\displaystyle E}
614:{\displaystyle V}
491:{\displaystyle y}
471:{\displaystyle x}
447:{\displaystyle y}
427:{\displaystyle x}
400:{\displaystyle y}
380:{\displaystyle x}
278:
199:{\displaystyle V}
94:undirected graphs
8013:
7911:
7904:
7897:
7888:
7887:
7874:
7873:
7864:
7863:
7854:
7853:
7674:Cross-validation
7646:Machine learning
7530:Social computing
7497:Network security
7292:Algorithm design
7221:Programming team
7181:Control variable
7158:Software library
7096:Software quality
7091:Operating system
7040:Network protocol
6905:Computer science
6898:
6891:
6884:
6875:
6874:
6859:
6858:
6847:
6846:
6835:
6834:
6824:
6823:
6755:Computer algebra
6730:Computer science
6450:Complex analysis
6284:
6277:
6270:
6261:
6260:
6237:Online textbooks
6155:
6130:
6109:
6100:
6087:
6078:
6069:
6056:
6043:
6041:
6024:
6012:
6000:
5989:
5980:
5961:
5950:
5941:
5922:
5921:
5901:
5895:
5894:
5892:
5872:
5866:
5865:
5863:
5843:
5837:
5836:
5834:
5816:
5807:
5801:
5798:
5792:
5791:
5790:
5789:
5783:
5776:
5762:
5756:
5755:
5744:
5738:
5737:
5736:
5735:
5709:
5703:
5702:
5700:
5698:10.1038/017284a0
5668:
5662:
5661:
5652:(2): 1056â1059,
5637:
5631:
5630:
5601:
5590:
5584:
5583:
5571:
5565:
5564:
5554:
5548:
5547:
5537:
5531:
5530:
5520:
5495:(7): 1955â1972.
5480:
5474:
5473:
5455:
5445:
5419:
5410:
5404:
5403:
5385:
5379:
5368:
5362:
5361:
5359:
5358:
5352:
5345:
5334:
5328:
5327:
5291:
5285:
5284:
5282:
5272:
5266:
5265:
5229:
5223:
5222:
5186:
5180:
5179:
5169:
5151:
5142:
5136:
5135:
5109:
5101:
5095:
5094:
5084:
5059:(7): 1955â1972.
5044:
5038:
5037:
5011:
5009:cond-mat/0304207
4991:
4985:
4984:
4948:
4932:
4926:
4920:
4911:
4908:
4902:
4896:
4890:
4883:
4877:
4871:
4855:Whitney, Hassler
4830:Szemerédi, Endre
4760:Golumbic, Martin
4592:Prim's algorithm
4495:Symmetric graphs
4365:Entitative graph
4345:Conceptual graph
4211:
4204:
3957:of the 5-vertex
3850:Wagner's Theorem
3793:graph properties
3750:of a graph. The
3736:Laplacian matrix
3728:adjacency matrix
3724:incidence matrix
3674:adjacency matrix
3580:Gustav Kirchhoff
3176:chemical editors
3022:computer science
3016:Computer science
2985:
2983:
2982:
2977:
2965:
2963:
2962:
2957:
2941:
2939:
2938:
2933:
2921:
2919:
2918:
2913:
2902:, its endpoints
2901:
2899:
2898:
2893:
2869:
2867:
2866:
2861:
2845:
2843:
2842:
2837:
2821:
2819:
2818:
2813:
2799:) respectively.
2784:
2782:
2781:
2776:
2774:
2770:
2769:
2768:
2699:
2697:
2696:
2691:
2679:
2677:
2676:
2671:
2669:
2665:
2664:
2663:
2600:
2598:
2597:
2592:
2580:
2578:
2577:
2572:
2570:
2566:
2555:
2554:
2551:
2547:
2546:
2490:which is not in
2489:
2487:
2486:
2481:
2457:
2455:
2454:
2449:
2412:
2410:
2409:
2404:
2402:
2398:
2387:
2386:
2383:
2379:
2378:
2279:
2277:
2276:
2271:
2241:
2239:
2238:
2233:
2218:
2216:
2215:
2210:
2161:
2159:
2158:
2153:
2125:
2123:
2122:
2117:
2093:
2091:
2090:
2085:
2073:
2071:
2070:
2065:
2049:
2047:
2046:
2041:
2029:
2027:
2026:
2021:
2002:
2000:
1999:
1994:
1983:of the edge and
1978:
1976:
1975:
1970:
1954:
1952:
1951:
1946:
1934:
1932:
1931:
1926:
1914:
1912:
1911:
1906:
1894:
1892:
1891:
1886:
1874:
1872:
1871:
1866:
1839:
1837:
1836:
1831:
1819:
1817:
1816:
1811:
1792:
1788:
1786:
1785:
1780:
1778:
1777:
1721:
1719:
1718:
1713:
1711:
1707:
1696:
1695:
1692:
1688:
1687:
1606:
1604:
1603:
1598:
1583:
1581:
1580:
1575:
1508:
1506:
1505:
1500:
1478:
1476:
1475:
1470:
1458:
1456:
1455:
1450:
1439:, its endpoints
1438:
1436:
1435:
1430:
1406:
1404:
1403:
1398:
1382:
1380:
1379:
1374:
1362:
1360:
1359:
1354:
1339:
1337:
1336:
1331:
1316:
1314:
1313:
1310:
1307:
1293:
1267:
1265:
1264:
1259:
1257:
1249:
1233:
1231:
1230:
1225:
1223:
1215:
1199:
1197:
1196:
1191:
1179:
1177:
1176:
1171:
1155:
1153:
1152:
1147:
1135:
1133:
1132:
1127:
1107:(sometimes also
1098:
1096:
1095:
1090:
1030:
1028:
1027:
1022:
1010:
1008:
1007:
1002:
948:
946:
945:
940:
928:
926:
925:
920:
905:
904:
901:
855:which is not in
854:
852:
851:
846:
810:
808:
807:
802:
762:
760:
759:
754:
739:
738:
735:
658:
656:
655:
650:
620:
618:
617:
612:
597:
595:
594:
589:
535:
523:
497:
495:
494:
489:
477:
475:
474:
469:
453:
451:
450:
445:
433:
431:
430:
425:
406:
404:
403:
398:
386:
384:
383:
378:
366:
364:
363:
358:
303:
301:
300:
295:
280:
279:
276:
205:
203:
202:
197:
182:
180:
179:
174:
50:is the study of
8021:
8020:
8016:
8015:
8014:
8012:
8011:
8010:
7996:
7995:
7994:
7989:
7966:
7918:
7915:
7885:
7880:
7871:
7842:
7823:Word processing
7731:
7717:Virtual reality
7678:
7640:
7611:Computer vision
7587:
7583:Multiprocessing
7549:
7511:
7477:Security hacker
7453:
7429:Digital library
7370:
7321:Mathematics of
7316:
7278:
7254:Automata theory
7249:Formal language
7230:
7196:Software design
7167:
7100:
7086:Virtual machine
7064:
7060:Network service
7021:
7012:Embedded system
6985:
6918:
6907:
6902:
6872:
6867:
6818:
6809:
6759:
6716:
6695:Systems science
6626:
6622:Homotopy theory
6588:
6555:
6507:
6479:
6426:
6373:
6344:Category theory
6330:
6295:
6288:
6239:
6230:Wayback Machine
6213:in your library
6180:Wayback Machine
6166:Wayback Machine
6140:
6137:
6127:
6039:
6033:
6022:
6009:
5977:
5949:. Paris: Dunod.
5930:
5925:
5918:
5902:
5898:
5873:
5869:
5844:
5840:
5814:
5808:
5804:
5799:
5795:
5787:
5785:
5781:
5774:
5763:
5759:
5748:Gardner, Martin
5745:
5741:
5733:
5731:
5729:
5710:
5706:
5669:
5665:
5638:
5634:
5628:
5612:(85): 172â176,
5591:
5587:
5572:
5568:
5555:
5551:
5538:
5534:
5481:
5477:
5417:
5411:
5407:
5400:
5386:
5382:
5369:
5365:
5356:
5354:
5350:
5343:
5335:
5331:
5292:
5288:
5273:
5269:
5230:
5226:
5187:
5183:
5149:
5143:
5139:
5102:
5098:
5045:
5041:
4992:
4988:
4973:
4933:
4929:
4921:
4914:
4909:
4905:
4897:
4893:
4884:
4880:
4872:
4868:
4864:
4859:
4815:Robertson, Neil
4810:Ringel, Gerhard
4795:Murty, U. S. R.
4745:Euler, Leonhard
4691:
4674:
4672:Generalizations
4647:
4615:
4513:
4410:Graph rewriting
4331:
4298:
4289:representations
4255:
4206:
4199:
4197:
4186:
4150:
4138:
4125:, for example:
4119:
4070:
4043:
3995:
3989:
3975:. For example:
3966:
3932:
3922:
3901:. For example,
3880:
3869:
3833:independent set
3803:. For example:
3781:
3769:
3764:
3752:distance matrix
3688:
3682:
3666:circle packings
3655:crossing number
3636:
3630:
3621:
3532:Heinrich Heesch
3471:Francis Guthrie
3322:
3314:weighted graphs
3310:
3290:
3250:
3237:rumor spreading
3210:
3208:Social sciences
3197:molecular graph
3144:
3076:
3058:geared towards
3056:graph databases
3018:
3010:network science
2992:
2971:
2968:
2967:
2951:
2948:
2947:
2942:are said to be
2927:
2924:
2923:
2907:
2904:
2903:
2875:
2872:
2871:
2855:
2852:
2851:
2831:
2828:
2827:
2807:
2804:
2803:
2764:
2760:
2723:
2719:
2705:
2702:
2701:
2685:
2682:
2681:
2659:
2655:
2618:
2614:
2606:
2603:
2602:
2586:
2583:
2582:
2550:
2549:
2542:
2538:
2501:
2497:
2495:
2492:
2491:
2463:
2460:
2459:
2443:
2440:
2439:
2382:
2381:
2374:
2370:
2333:
2329:
2315:
2312:
2311:
2265:
2262:
2261:
2227:
2224:
2223:
2180:
2177:
2176:
2135:
2132:
2131:
2099:
2096:
2095:
2079:
2076:
2075:
2059:
2056:
2055:
2035:
2032:
2031:
2015:
2012:
2011:
1988:
1985:
1984:
1964:
1961:
1960:
1955:are called the
1940:
1937:
1936:
1920:
1917:
1916:
1915:, the vertices
1900:
1897:
1896:
1880:
1877:
1876:
1848:
1845:
1844:
1825:
1822:
1821:
1802:
1799:
1798:
1797:of elements of
1790:
1773:
1769:
1767:
1764:
1763:
1691:
1690:
1683:
1679:
1642:
1638:
1630:
1627:
1626:
1592:
1589:
1588:
1551:
1548:
1547:
1521:
1515:
1488:
1485:
1484:
1479:are said to be
1464:
1461:
1460:
1444:
1441:
1440:
1412:
1409:
1408:
1392:
1389:
1388:
1368:
1365:
1364:
1348:
1345:
1344:
1325:
1322:
1321:
1311:
1308:
1298:
1297:
1295:
1288:
1253:
1245:
1243:
1240:
1239:
1219:
1211:
1209:
1206:
1205:
1185:
1182:
1181:
1165:
1162:
1161:
1141:
1138:
1137:
1121:
1118:
1117:
1036:
1033:
1032:
1016:
1013:
1012:
954:
951:
950:
934:
931:
930:
900:
899:
860:
857:
856:
816:
813:
812:
796:
793:
792:
734:
733:
682:
679:
678:
644:
641:
640:
606:
603:
602:
559:
556:
555:
548:
547:
546:
545:
541:
540:
539:
536:
528:
527:
524:
483:
480:
479:
463:
460:
459:
439:
436:
435:
419:
416:
415:
407:are called the
392:
389:
388:
372:
369:
368:
367:, the vertices
340:
337:
336:
322:unordered pairs
275:
274:
229:
226:
225:
191:
188:
187:
150:
147:
146:
128:
116:
110:
98:directed graphs
28:
25:
12:
11:
5:
8019:
8009:
8008:
7991:
7990:
7988:
7987:
7982:
7976:
7974:
7968:
7967:
7965:
7964:
7959:
7954:
7949:
7944:
7939:
7934:
7928:
7926:
7920:
7919:
7914:
7913:
7906:
7899:
7891:
7882:
7881:
7879:
7878:
7868:
7858:
7847:
7844:
7843:
7841:
7840:
7835:
7830:
7825:
7820:
7815:
7810:
7805:
7800:
7795:
7790:
7785:
7780:
7775:
7770:
7765:
7760:
7755:
7750:
7745:
7739:
7737:
7733:
7732:
7730:
7729:
7727:Solid modeling
7724:
7719:
7714:
7709:
7704:
7699:
7694:
7688:
7686:
7680:
7679:
7677:
7676:
7671:
7666:
7661:
7656:
7650:
7648:
7642:
7641:
7639:
7638:
7633:
7628:
7626:Control method
7623:
7618:
7613:
7608:
7603:
7597:
7595:
7589:
7588:
7586:
7585:
7580:
7578:Multithreading
7575:
7570:
7565:
7559:
7557:
7551:
7550:
7548:
7547:
7542:
7537:
7532:
7527:
7521:
7519:
7513:
7512:
7510:
7509:
7504:
7499:
7494:
7489:
7484:
7479:
7474:
7472:Formal methods
7469:
7463:
7461:
7455:
7454:
7452:
7451:
7446:
7444:World Wide Web
7441:
7436:
7431:
7426:
7421:
7416:
7411:
7406:
7401:
7396:
7391:
7386:
7380:
7378:
7372:
7371:
7369:
7368:
7363:
7358:
7353:
7348:
7343:
7338:
7333:
7327:
7325:
7318:
7317:
7315:
7314:
7309:
7304:
7299:
7294:
7288:
7286:
7280:
7279:
7277:
7276:
7271:
7266:
7261:
7256:
7251:
7246:
7240:
7238:
7232:
7231:
7229:
7228:
7223:
7218:
7213:
7208:
7203:
7198:
7193:
7188:
7183:
7177:
7175:
7169:
7168:
7166:
7165:
7160:
7155:
7150:
7145:
7140:
7135:
7130:
7125:
7120:
7114:
7112:
7102:
7101:
7099:
7098:
7093:
7088:
7083:
7078:
7072:
7070:
7066:
7065:
7063:
7062:
7057:
7052:
7047:
7042:
7037:
7031:
7029:
7023:
7022:
7020:
7019:
7014:
7009:
7004:
6999:
6993:
6991:
6987:
6986:
6984:
6983:
6974:
6969:
6964:
6959:
6954:
6949:
6944:
6939:
6934:
6928:
6926:
6920:
6919:
6912:
6909:
6908:
6901:
6900:
6893:
6886:
6878:
6869:
6868:
6866:
6865:
6853:
6841:
6829:
6814:
6811:
6810:
6808:
6807:
6802:
6797:
6792:
6787:
6782:
6781:
6780:
6773:Mathematicians
6769:
6767:
6765:Related topics
6761:
6760:
6758:
6757:
6752:
6747:
6742:
6737:
6732:
6726:
6724:
6718:
6717:
6715:
6714:
6713:
6712:
6707:
6702:
6700:Control theory
6692:
6687:
6682:
6677:
6672:
6667:
6662:
6657:
6652:
6647:
6642:
6636:
6634:
6628:
6627:
6625:
6624:
6619:
6614:
6609:
6604:
6598:
6596:
6590:
6589:
6587:
6586:
6581:
6576:
6571:
6565:
6563:
6557:
6556:
6554:
6553:
6548:
6543:
6538:
6533:
6528:
6523:
6517:
6515:
6509:
6508:
6506:
6505:
6500:
6495:
6489:
6487:
6481:
6480:
6478:
6477:
6475:Measure theory
6472:
6467:
6462:
6457:
6452:
6447:
6442:
6436:
6434:
6428:
6427:
6425:
6424:
6419:
6414:
6409:
6404:
6399:
6394:
6389:
6383:
6381:
6375:
6374:
6372:
6371:
6366:
6361:
6356:
6351:
6346:
6340:
6338:
6332:
6331:
6329:
6328:
6323:
6318:
6317:
6316:
6311:
6300:
6297:
6296:
6287:
6286:
6279:
6272:
6264:
6258:
6257:
6252:
6246:
6238:
6235:
6234:
6233:
6220:
6206:
6200:
6194:
6188:
6183:
6173:
6168:
6156:
6142:"Graph theory"
6136:
6135:External links
6133:
6132:
6131:
6125:
6110:
6101:
6088:
6079:
6070:
6066:Academic Press
6057:
6044:
6031:
6013:
6007:
5990:
5981:
5975:
5962:
5953:
5942:
5929:
5926:
5924:
5923:
5916:
5896:
5867:
5854:(3): 491â567,
5838:
5825:(3): 429â490,
5802:
5793:
5757:
5739:
5727:
5704:
5663:
5632:
5626:
5585:
5566:
5549:
5532:
5475:
5405:
5398:
5380:
5363:
5329:
5286:
5267:
5240:(2): 134â143.
5224:
5197:(2): 473â485.
5181:
5160:(1): 1171458.
5137:
5118:(5): 784â786.
5096:
5039:
5002:(1): 113â121.
4986:
4971:
4927:
4925:, p. 161.
4912:
4903:
4901:, p. 149.
4891:
4878:
4876:, p. 148.
4865:
4863:
4860:
4858:
4857:
4852:
4847:
4842:
4837:
4832:
4827:
4825:Sudakov, Benny
4822:
4817:
4812:
4807:
4802:
4797:
4792:
4790:LovĂĄsz, LĂĄszlĂł
4787:
4782:
4777:
4772:
4767:
4765:Graham, Ronald
4762:
4757:
4752:
4750:Faudree, Ralph
4747:
4742:
4737:
4732:
4727:
4722:
4717:
4712:
4707:
4705:BollobĂĄs, BĂ©la
4702:
4697:
4690:
4687:
4686:
4685:
4680:
4673:
4670:
4669:
4668:
4663:
4658:
4653:
4646:
4643:
4642:
4641:
4636:
4631:
4626:
4621:
4614:
4611:
4610:
4609:
4604:
4599:
4594:
4589:
4584:
4579:
4574:
4569:
4564:
4559:
4554:
4549:
4544:
4539:
4534:
4529:
4524:
4519:
4512:
4509:
4508:
4507:
4502:
4497:
4492:
4487:
4482:
4477:
4472:
4467:
4462:
4457:
4452:
4447:
4445:Network theory
4442:
4437:
4432:
4427:
4422:
4420:Graph property
4417:
4412:
4407:
4405:Graph equation
4402:
4397:
4392:
4390:Graph database
4387:
4385:Graph coloring
4382:
4377:
4372:
4367:
4362:
4357:
4352:
4350:Data structure
4347:
4342:
4340:Citation graph
4337:
4330:
4329:Related topics
4327:
4326:
4325:
4320:
4315:
4310:
4305:
4299:
4297:
4294:
4293:
4292:
4285:
4274:
4271:
4265:
4254:
4251:
4250:
4249:
4239:
4229:
4223:
4193:
4185:
4182:
4181:
4180:
4177:
4171:
4164:Dominating set
4149:
4146:
4145:
4144:
4137:
4134:
4133:
4132:
4118:
4115:
4114:
4113:
4107:
4102:
4097:
4092:
4087:
4081:
4076:
4069:
4068:Route problems
4066:
4042:
4039:
4038:
4037:
4031:
4025:
4020:, also called
4015:
4010:
4005:
3993:Graph coloring
3991:Main article:
3988:
3987:Graph coloring
3985:
3984:
3983:
3969:
3968:
3964:
3959:complete graph
3935:
3934:
3930:
3925:complete graph
3920:
3883:
3882:
3878:
3867:
3841:
3840:
3839:(NP-complete).
3835:is called the
3813:
3812:
3811:(NP-complete).
3809:clique problem
3780:
3777:
3768:
3765:
3763:
3760:
3748:spanning trees
3717:adjacency list
3692:data structure
3684:Main article:
3681:
3678:
3632:Main article:
3629:
3626:
3620:
3619:Representation
3617:
3540:Wolfgang Haken
3448:
3447:
3353:analysis situs
3347:knight problem
3334:Leonhard Euler
3321:
3318:
3309:
3306:
3289:
3286:
3249:
3246:
3235:or to explore
3209:
3206:
3143:
3140:
3138:, and others.
3120:lattice graphs
3075:
3072:
3017:
3014:
2991:
2988:
2975:
2955:
2931:
2911:
2891:
2888:
2885:
2882:
2879:
2859:
2835:
2811:
2773:
2767:
2763:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2722:
2718:
2715:
2712:
2709:
2689:
2668:
2662:
2658:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2617:
2613:
2610:
2590:
2569:
2565:
2562:
2559:
2545:
2541:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2504:
2500:
2479:
2476:
2473:
2470:
2467:
2447:
2423:
2422:
2401:
2397:
2394:
2391:
2377:
2373:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2332:
2328:
2325:
2322:
2319:
2309:
2298:directed lines
2294:directed links
2290:directed edges
2269:
2259:
2231:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2173:directed graph
2165:Multiple edges
2151:
2148:
2145:
2142:
2139:
2126:is called the
2115:
2112:
2109:
2106:
2103:
2083:
2063:
2039:
2019:
1992:
1968:
1944:
1924:
1904:
1884:
1875:directed from
1864:
1861:
1858:
1855:
1852:
1829:
1809:
1806:
1776:
1772:
1756:
1755:
1740:directed lines
1736:directed links
1732:directed edges
1710:
1706:
1703:
1700:
1686:
1682:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1641:
1637:
1634:
1624:
1596:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1544:directed graph
1533:directed graph
1519:Directed graph
1517:Main article:
1514:
1513:Directed graph
1511:
1498:
1495:
1492:
1468:
1448:
1428:
1425:
1422:
1419:
1416:
1396:
1372:
1352:
1329:
1256:
1252:
1248:
1238:of a graph is
1222:
1218:
1214:
1204:of a graph is
1189:
1169:
1145:
1125:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1020:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
938:
918:
915:
912:
909:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
844:
841:
838:
835:
832:
829:
826:
823:
820:
800:
773:
772:
769:unordered pair
752:
749:
746:
743:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
676:
648:
638:
610:
587:
584:
581:
578:
575:
572:
569:
566:
563:
543:
542:
537:
530:
529:
525:
518:
517:
516:
515:
514:
501:multiple edges
487:
467:
443:
423:
396:
376:
356:
353:
350:
347:
344:
326:
325:
293:
290:
287:
284:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
223:
195:
172:
169:
166:
163:
160:
157:
154:
127:
124:
109:
106:
26:
9:
6:
4:
3:
2:
8018:
8007:
8004:
8003:
8001:
7986:
7983:
7981:
7978:
7977:
7975:
7973:
7969:
7963:
7960:
7958:
7955:
7953:
7950:
7948:
7945:
7943:
7940:
7938:
7935:
7933:
7930:
7929:
7927:
7925:
7921:
7912:
7907:
7905:
7900:
7898:
7893:
7892:
7889:
7877:
7869:
7867:
7859:
7857:
7849:
7848:
7845:
7839:
7836:
7834:
7831:
7829:
7826:
7824:
7821:
7819:
7816:
7814:
7811:
7809:
7806:
7804:
7801:
7799:
7796:
7794:
7791:
7789:
7786:
7784:
7781:
7779:
7776:
7774:
7771:
7769:
7766:
7764:
7761:
7759:
7756:
7754:
7751:
7749:
7746:
7744:
7741:
7740:
7738:
7734:
7728:
7725:
7723:
7720:
7718:
7715:
7713:
7712:Mixed reality
7710:
7708:
7705:
7703:
7700:
7698:
7695:
7693:
7690:
7689:
7687:
7685:
7681:
7675:
7672:
7670:
7667:
7665:
7662:
7660:
7657:
7655:
7652:
7651:
7649:
7647:
7643:
7637:
7634:
7632:
7629:
7627:
7624:
7622:
7619:
7617:
7614:
7612:
7609:
7607:
7604:
7602:
7599:
7598:
7596:
7594:
7590:
7584:
7581:
7579:
7576:
7574:
7571:
7569:
7566:
7564:
7561:
7560:
7558:
7556:
7552:
7546:
7545:Accessibility
7543:
7541:
7540:Visualization
7538:
7536:
7533:
7531:
7528:
7526:
7523:
7522:
7520:
7518:
7514:
7508:
7505:
7503:
7500:
7498:
7495:
7493:
7490:
7488:
7485:
7483:
7480:
7478:
7475:
7473:
7470:
7468:
7465:
7464:
7462:
7460:
7456:
7450:
7447:
7445:
7442:
7440:
7437:
7435:
7432:
7430:
7427:
7425:
7422:
7420:
7417:
7415:
7412:
7410:
7407:
7405:
7402:
7400:
7397:
7395:
7392:
7390:
7387:
7385:
7382:
7381:
7379:
7377:
7373:
7367:
7364:
7362:
7359:
7357:
7354:
7352:
7349:
7347:
7344:
7342:
7339:
7337:
7334:
7332:
7329:
7328:
7326:
7324:
7319:
7313:
7310:
7308:
7305:
7303:
7300:
7298:
7295:
7293:
7290:
7289:
7287:
7285:
7281:
7275:
7272:
7270:
7267:
7265:
7262:
7260:
7257:
7255:
7252:
7250:
7247:
7245:
7242:
7241:
7239:
7237:
7233:
7227:
7224:
7222:
7219:
7217:
7214:
7212:
7209:
7207:
7204:
7202:
7199:
7197:
7194:
7192:
7189:
7187:
7184:
7182:
7179:
7178:
7176:
7174:
7170:
7164:
7161:
7159:
7156:
7154:
7151:
7149:
7146:
7144:
7141:
7139:
7136:
7134:
7131:
7129:
7126:
7124:
7121:
7119:
7116:
7115:
7113:
7111:
7107:
7103:
7097:
7094:
7092:
7089:
7087:
7084:
7082:
7079:
7077:
7074:
7073:
7071:
7067:
7061:
7058:
7056:
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7036:
7033:
7032:
7030:
7028:
7024:
7018:
7015:
7013:
7010:
7008:
7007:Dependability
7005:
7003:
7000:
6998:
6995:
6994:
6992:
6988:
6982:
6978:
6975:
6973:
6970:
6968:
6965:
6963:
6960:
6958:
6955:
6953:
6950:
6948:
6945:
6943:
6940:
6938:
6935:
6933:
6930:
6929:
6927:
6925:
6921:
6916:
6910:
6906:
6899:
6894:
6892:
6887:
6885:
6880:
6879:
6876:
6864:
6863:
6854:
6852:
6851:
6842:
6840:
6839:
6830:
6828:
6827:
6822:
6816:
6815:
6812:
6806:
6803:
6801:
6798:
6796:
6793:
6791:
6788:
6786:
6783:
6779:
6776:
6775:
6774:
6771:
6770:
6768:
6766:
6762:
6756:
6753:
6751:
6748:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6728:
6727:
6725:
6723:
6722:Computational
6719:
6711:
6708:
6706:
6703:
6701:
6698:
6697:
6696:
6693:
6691:
6688:
6686:
6683:
6681:
6678:
6676:
6673:
6671:
6668:
6666:
6663:
6661:
6658:
6656:
6653:
6651:
6648:
6646:
6643:
6641:
6638:
6637:
6635:
6633:
6629:
6623:
6620:
6618:
6615:
6613:
6610:
6608:
6605:
6603:
6600:
6599:
6597:
6595:
6591:
6585:
6582:
6580:
6577:
6575:
6572:
6570:
6567:
6566:
6564:
6562:
6561:Number theory
6558:
6552:
6549:
6547:
6544:
6542:
6539:
6537:
6534:
6532:
6529:
6527:
6524:
6522:
6519:
6518:
6516:
6514:
6510:
6504:
6501:
6499:
6496:
6494:
6493:Combinatorics
6491:
6490:
6488:
6486:
6482:
6476:
6473:
6471:
6468:
6466:
6463:
6461:
6458:
6456:
6453:
6451:
6448:
6446:
6445:Real analysis
6443:
6441:
6438:
6437:
6435:
6433:
6429:
6423:
6420:
6418:
6415:
6413:
6410:
6408:
6405:
6403:
6400:
6398:
6395:
6393:
6390:
6388:
6385:
6384:
6382:
6380:
6376:
6370:
6367:
6365:
6362:
6360:
6357:
6355:
6352:
6350:
6347:
6345:
6342:
6341:
6339:
6337:
6333:
6327:
6324:
6322:
6319:
6315:
6312:
6310:
6307:
6306:
6305:
6302:
6301:
6298:
6293:
6285:
6280:
6278:
6273:
6271:
6266:
6265:
6262:
6256:
6253:
6250:
6247:
6244:
6241:
6240:
6231:
6227:
6224:
6221:
6218:
6214:
6210:
6207:
6204:
6201:
6198:
6195:
6192:
6189:
6187:
6184:
6181:
6177:
6174:
6172:
6169:
6167:
6163:
6160:
6157:
6153:
6149:
6148:
6143:
6139:
6138:
6128:
6122:
6118:
6117:
6111:
6107:
6102:
6098:
6097:North-Holland
6094:
6089:
6085:
6080:
6076:
6071:
6067:
6063:
6058:
6054:
6050:
6045:
6038:
6034:
6032:0-13-363473-6
6028:
6021:
6020:
6014:
6010:
6008:0-486-24775-9
6004:
5999:
5998:
5991:
5987:
5982:
5978:
5972:
5968:
5963:
5959:
5954:
5948:
5943:
5939:
5938:
5932:
5931:
5919:
5913:
5909:
5908:
5900:
5891:
5886:
5882:
5878:
5871:
5862:
5857:
5853:
5849:
5842:
5833:
5828:
5824:
5820:
5813:
5806:
5797:
5784:on 2016-03-05
5780:
5773:
5772:
5767:
5761:
5753:
5749:
5743:
5730:
5724:
5720:
5719:
5714:
5708:
5699:
5694:
5690:
5686:
5682:
5678:
5674:
5667:
5659:
5655:
5651:
5647:
5643:
5636:
5629:
5627:9780511703690
5623:
5619:
5615:
5611:
5608:, Series IV,
5607:
5606:
5600:
5595:
5589:
5581:
5577:
5570:
5562:
5561:
5553:
5545:
5544:
5536:
5528:
5524:
5519:
5514:
5510:
5506:
5502:
5498:
5494:
5490:
5486:
5479:
5471:
5467:
5463:
5459:
5454:
5449:
5444:
5439:
5435:
5431:
5427:
5423:
5416:
5409:
5401:
5395:
5391:
5384:
5377:
5373:
5367:
5353:on 2020-07-28
5349:
5342:
5341:
5333:
5325:
5321:
5317:
5313:
5309:
5305:
5302:(1): 015102.
5301:
5297:
5290:
5281:
5280:
5271:
5263:
5259:
5255:
5251:
5247:
5243:
5239:
5235:
5228:
5220:
5216:
5212:
5208:
5204:
5200:
5196:
5192:
5185:
5177:
5173:
5168:
5163:
5159:
5155:
5148:
5141:
5133:
5129:
5125:
5121:
5117:
5113:
5108:
5100:
5092:
5088:
5083:
5078:
5074:
5070:
5066:
5062:
5058:
5054:
5050:
5043:
5035:
5031:
5027:
5023:
5019:
5015:
5010:
5005:
5001:
4997:
4990:
4982:
4978:
4974:
4972:9781450326223
4968:
4964:
4960:
4956:
4952:
4947:
4942:
4938:
4931:
4924:
4919:
4917:
4907:
4900:
4895:
4888:
4882:
4875:
4870:
4866:
4856:
4853:
4851:
4848:
4846:
4843:
4841:
4838:
4836:
4835:Thomas, Robin
4833:
4831:
4828:
4826:
4823:
4821:
4820:Seymour, Paul
4818:
4816:
4813:
4811:
4808:
4806:
4805:Rényi, Alfréd
4803:
4801:
4798:
4796:
4793:
4791:
4788:
4786:
4783:
4781:
4780:Kotzig, Anton
4778:
4776:
4773:
4771:
4770:Harary, Frank
4768:
4766:
4763:
4761:
4758:
4756:
4753:
4751:
4748:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4706:
4703:
4701:
4700:Berge, Claude
4698:
4696:
4693:
4692:
4684:
4681:
4679:
4676:
4675:
4667:
4666:Ramsey theory
4664:
4662:
4659:
4657:
4654:
4652:
4651:Combinatorics
4649:
4648:
4640:
4637:
4635:
4632:
4630:
4627:
4625:
4622:
4620:
4617:
4616:
4608:
4605:
4603:
4600:
4598:
4595:
4593:
4590:
4588:
4585:
4583:
4580:
4578:
4575:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4543:
4540:
4538:
4535:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4515:
4514:
4506:
4503:
4501:
4498:
4496:
4493:
4491:
4488:
4486:
4483:
4481:
4478:
4476:
4473:
4471:
4470:Quantum graph
4468:
4466:
4465:Perfect graph
4463:
4461:
4458:
4456:
4453:
4451:
4448:
4446:
4443:
4441:
4438:
4436:
4435:Logical graph
4433:
4431:
4430:Knight's Tour
4428:
4426:
4423:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4400:Graph drawing
4398:
4396:
4393:
4391:
4388:
4386:
4383:
4381:
4378:
4376:
4375:Graph algebra
4373:
4371:
4368:
4366:
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4346:
4343:
4341:
4338:
4336:
4333:
4332:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4300:
4290:
4286:
4283:
4279:
4275:
4272:
4270:
4266:
4263:
4260:
4259:
4258:
4253:Graph classes
4247:
4246:regular graph
4243:
4240:
4237:
4233:
4232:Edge coloring
4230:
4227:
4224:
4221:
4218:
4217:
4216:
4213:
4209:
4202:
4196:
4192:
4178:
4175:
4172:
4169:
4168:neighborhoods
4165:
4162:
4161:
4160:
4158:
4154:
4143:
4140:
4139:
4131:
4128:
4127:
4126:
4124:
4111:
4108:
4106:
4103:
4101:
4098:
4096:
4093:
4091:
4088:
4085:
4082:
4080:
4077:
4075:
4072:
4071:
4065:
4063:
4059:
4055:
4054:compositional
4050:
4048:
4047:partial order
4035:
4032:
4029:
4026:
4023:
4019:
4016:
4014:
4011:
4009:
4006:
4004:
4001:
4000:
3999:
3994:
3982:
3978:
3977:
3976:
3974:
3963:
3960:
3956:
3952:
3948:
3944:
3943:
3942:
3940:
3929:
3926:
3919:
3916:
3912:
3908:
3907:
3906:
3904:
3900:
3896:
3895:homeomorphism
3892:
3888:
3877:
3873:
3866:
3863:
3859:
3855:
3854:
3853:
3851:
3846:
3838:
3834:
3830:
3829:
3828:
3825:
3820:
3818:
3810:
3806:
3805:
3804:
3802:
3798:
3794:
3790:
3786:
3776:
3774:
3759:
3757:
3756:shortest path
3753:
3749:
3745:
3741:
3737:
3733:
3732:degree matrix
3729:
3725:
3720:
3718:
3714:
3709:
3706:
3702:
3701:sparse graphs
3697:
3693:
3687:
3677:
3675:
3671:
3667:
3662:
3660:
3656:
3651:
3649:
3644:
3640:
3635:
3634:Graph drawing
3625:
3616:
3614:
3613:
3608:
3604:
3599:
3597:
3593:
3589:
3585:
3581:
3577:
3573:
3569:
3565:
3560:
3558:
3554:
3550:
3546:
3541:
3537:
3536:Kenneth Appel
3533:
3528:
3526:
3525:
3520:
3516:
3512:
3508:
3504:
3500:
3496:
3492:
3488:
3484:
3480:
3477:addressed to
3476:
3472:
3468:
3463:
3461:
3457:
3453:
3445:
3441:
3437:
3433:
3429:
3428:
3427:
3425:
3424:
3419:
3414:
3411:
3407:
3403:
3399:
3395:
3394:
3389:
3385:
3381:
3377:
3372:
3370:
3366:
3362:
3358:
3355:initiated by
3354:
3350:
3348:
3343:
3339:
3335:
3326:
3317:
3315:
3305:
3303:
3299:
3295:
3285:
3283:
3278:
3275:
3271:
3267:
3263:
3258:
3255:
3245:
3242:
3238:
3234:
3230:
3222:
3219:
3214:
3205:
3203:
3198:
3194:
3190:
3185:
3181:
3177:
3173:
3169:
3165:
3161:
3157:
3153:
3149:
3139:
3137:
3133:
3129:
3125:
3121:
3118:, which uses
3117:
3113:
3109:
3105:
3101:
3097:
3093:
3089:
3085:
3081:
3071:
3069:
3065:
3061:
3057:
3053:
3049:
3045:
3041:
3036:
3032:
3027:
3023:
3013:
3011:
3006:
2996:
2987:
2973:
2953:
2945:
2929:
2909:
2886:
2883:
2880:
2857:
2849:
2833:
2825:
2809:
2800:
2798:
2797:
2792:
2788:
2771:
2765:
2761:
2757:
2751:
2748:
2745:
2739:
2733:
2730:
2727:
2720:
2713:
2710:
2707:
2687:
2666:
2660:
2656:
2652:
2646:
2643:
2640:
2634:
2628:
2625:
2622:
2615:
2611:
2608:
2588:
2567:
2563:
2560:
2557:
2543:
2539:
2535:
2529:
2526:
2523:
2517:
2511:
2508:
2505:
2498:
2474:
2471:
2468:
2445:
2437:
2436:
2430:
2428:
2420:
2416:
2399:
2395:
2392:
2389:
2375:
2371:
2367:
2361:
2358:
2355:
2349:
2343:
2340:
2337:
2330:
2323:
2320:
2317:
2310:
2307:
2303:
2299:
2295:
2291:
2288:(also called
2287:
2283:
2267:
2260:
2257:
2253:
2250:(also called
2249:
2245:
2229:
2222:
2221:
2220:
2203:
2200:
2197:
2194:
2191:
2185:
2182:
2174:
2169:
2167:
2166:
2146:
2143:
2140:
2129:
2128:inverted edge
2110:
2107:
2104:
2081:
2061:
2053:
2037:
2017:
2010:
2006:
1990:
1982:
1966:
1959:of the edge,
1958:
1942:
1922:
1902:
1882:
1859:
1856:
1853:
1841:
1827:
1807:
1804:
1796:
1774:
1770:
1761:
1753:
1752:ordered pairs
1749:
1745:
1741:
1737:
1733:
1730:(also called
1729:
1725:
1708:
1704:
1701:
1698:
1684:
1680:
1676:
1670:
1667:
1664:
1658:
1652:
1649:
1646:
1639:
1635:
1632:
1625:
1622:
1618:
1615:(also called
1614:
1610:
1594:
1587:
1586:
1585:
1568:
1565:
1562:
1556:
1553:
1545:
1540:
1538:
1534:
1525:
1520:
1510:
1496:
1493:
1490:
1482:
1466:
1446:
1423:
1420:
1417:
1394:
1386:
1370:
1350:
1343:
1327:
1318:
1305:
1301:
1291:
1286:
1281:
1279:
1275:
1271:
1250:
1237:
1216:
1203:
1187:
1167:
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7972:Proprietary
7818:Video games
7798:Digital art
7555:Concurrency
7424:Data mining
7336:Probability
7076:Interpreter
6862:WikiProject
6705:Game theory
6685:Probability
6422:Homological
6412:Multilinear
6392:Commutative
6369:Type theory
6336:Foundations
6292:mathematics
5713:Tutte, W.T.
4740:ErdĆs, Paul
4661:Knot theory
4262:Enumerating
4238:as possible
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3648:W. T. Tutte
3460:PĂłlya Prize
3452:DĂ©nes KĆnig
3342:Vandermonde
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3080:linguistics
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777:undirected
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7748:E-commerce
7341:Statistics
7284:Algorithms
7081:Middleware
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6690:Statistics
6569:Arithmetic
6531:Arithmetic
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6364:Set theory
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5788:2016-03-14
5734:2016-03-14
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4278:algorithms
4220:Arboricity
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3797:hereditary
3568:Kuratowski
3378:and while
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3170:and edges
3162:summarize
3128:TextGraphs
3102:. Within
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3040:algorithms
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779:multigraph
454:and to be
7932:Cytoscape
7697:Rendering
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7323:computing
7274:Semantics
6972:Processor
6617:Geometric
6607:Algebraic
6546:Euclidean
6521:Algebraic
6417:Universal
6152:EMS Press
6001:. Dover.
5509:0006-8950
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5462:2475-9066
5324:0021-8979
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3696:algorithm
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7128:Compiler
7027:Networks
6924:Hardware
6838:Category
6594:Topology
6541:Discrete
6526:Analytic
6513:Geometry
6485:Discrete
6440:Calculus
6432:Analysis
6387:Abstract
6326:Glossary
6309:Timeline
6226:Archived
6162:Archived
6037:Archived
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627:vertices
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6632:Applied
6602:General
6379:Algebra
6304:History
6178:at the
6154:, 2001
5685:Bibcode
5518:6598625
5438:bioRxiv
5430:Bibcode
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4951:Bibcode
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3132:WordNet
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1075:,
1072:x
1066:}
1063:y
1060:,
1057:x
1054:{
1051:{
1045:E
1042::
999:}
996:V
990:y
987:,
984:x
978:}
975:y
972:,
969:x
966:{
963:{
957:E
937:E
917:}
914:y
908:x
896:V
890:y
887:,
884:x
878:}
875:y
872:,
869:x
866:{
863:{
843:}
840:x
837:{
834:=
831:}
828:x
825:,
822:x
819:{
799:x
751:}
748:y
742:x
730:V
724:y
721:,
718:x
712:}
709:y
706:,
703:x
700:{
697:{
691:E
688::
647:E
609:V
586:)
580:,
577:E
574:,
571:V
568:(
565:=
562:G
486:y
466:x
442:y
422:x
395:y
375:x
355:}
352:y
349:,
346:x
343:{
292:}
289:y
283:x
271:V
265:y
262:,
259:x
253:}
250:y
247:,
244:x
241:{
238:{
232:E
194:V
171:)
168:E
165:,
162:V
159:(
156:=
153:G
24:.
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