4060:. In these applications, graphs are ordered by specificity, meaning that more constrained graphs—which are more specific and thus contain a greater amount of information—are 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.
3006:
519:
544:
532:
42:
6832:
142:
1535:
3327:, 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.
3224:
7870:
6844:
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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
3653:
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
3634:
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,
3553:
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
3267:
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
3210:
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
3254:
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
3649:
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.
4008:
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
3197:
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
3837:
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:
3718:
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.
3336:
3469:, 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
3858:
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,
3287:
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.
3013:
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
2422:
1731:
3423:
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.
3198:
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
5962:
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
3048:
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
2590:
3709:
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
772:
313:
3169:, 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
2794:
2689:
938:
3039:
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
3480:: "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
3018:
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
5822:
4200:
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
1108:
1020:
5157:
3668:
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
3635:
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.
4612:
2228:
864:
607:
3810:
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
7926:
1167:
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
3177:
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
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376:
192:
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2499:
2171:
2135:
1884:
1798:
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The network graph formed by
Knowledge (XXG) editors (edges) contributing to different Knowledge (XXG) language versions (vertices) during one month in summer 2013.
6292:
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2324:
1639:
1277:
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2879:
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2103:
2083:
2059:
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2012:
1988:
1964:
1944:
1924:
1904:
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1616:
1488:
1468:
1416:
1392:
1349:
1209:
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1165:
1145:
958:
820:
668:
630:
507:
487:
463:
443:
416:
396:
215:
3765:, 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
3741:, 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
5349:
5494:
Shah, Preya; Ashourvan, Arian; Mikhail, Fadi; Pines, Adam; Kini, Lohith; Oechsel, Kelly; Das, Sandhitsu R; Stein, Joel M; Shinohara, Russell T (2019-07-01).
5200:
Vecchio, F (2017). ""Small World" architecture in brain connectivity and hippocampal volume in
Alzheimer's disease: a study via graph theory from EEG data".
5058:
Shah, Preya; Ashourvan, Arian; Mikhail, Fadi; Pines, Adam; Kini, Lohith; Oechsel, Kelly; Das, Sandhitsu R; Stein, Joel M; Shinohara, Russell T (2019-07-01).
4023:
3193:, 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
3661:
was very influential on the subject of graph drawing. Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings.
3654:
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.
6906:
3279:
to model and analyse datasets with complex relationships. For example, graph-based methods are often used to 'cluster' cells together into cell-types in
3137:) 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
2504:
4328:
2449:
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
4607:
3830:. 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.
691:
238:
4009:
coincident edges are the same color), or other variations. Among the famous results and conjectures concerning graph coloring are the following:
7623:
7595:
3701:
The tabular representation lends itself well to computational applications. There are different ways to store graphs in a computer system. The
7919:
7648:
6135:
5985:
5926:
5737:
5408:
3730:, 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.
802:
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
7499:
6655:
5305:
Kumar, Ankush; Kulkarni, G. U. (2016-01-04). "Evaluating conducting network based transparent electrodes from geometrical considerations".
3102:
6872:
5653:"Ueber die Analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen"
7653:
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7805:
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with particular properties. Enumerative graph theory then arose from the results of Cayley and the fundamental results published by
6236:
3982:
Another class of problems has to do with the extent to which various species and generalizations of graphs are determined by their
3908:
of a graph is any graph obtained by subdividing some (or no) edges. Subdivision containment is related to graph properties such as
7969:
7912:
7893:
6987:
6336:
6107:
4279:
3117:, especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words;
7281:
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3375:
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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
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6899:
6848:
4825:
3843:
3665:
3555:
3247:
3074:
87:
5997:
Mathematical results on scale-free random graphs in "Handbook of Graphs and
Networks" (S. Bornholdt and H.G. Schuster (eds))
4100:
3589:
came from the use of the techniques of modern algebra. The first example of such a use comes from the work of the physicist
3348:
7714:
7691:
7421:
7411:
6278:
6162:
5570:
4268:
Many problems involve characterizing the members of various classes of graphs. Some examples of such questions are below:
4064:
3370:. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by
3098:
4562:
4557:
1045:
963:
7795:
7383:
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7196:
6972:
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6172:
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4018:
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3559:
63:
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3786:: the problem of counting graphs meeting specified conditions. Some of this work is found in Harary and Palmer (1973).
3005:
7883:
7618:
7116:
6775:
6331:
6157:
5425:
4693:
4587:
4365:
4323:
3295:; nervous systems can be seen as a graph, where the nodes are neurons and the edges are the connections between them.
518:
6186:
4552:
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3594:
1538:
A directed graph with three vertices and four directed edges (the double arrow represents an edge in each direction).
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Still another such problem, the minor containment problem, is to find a fixed graph as a minor of a given graph. A
3696:
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3194:
6196:
7873:
7800:
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7286:
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7724:
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7143:
7012:
6805:
4649:
4318:
4140:
4094:
4084:
4068:
4067:, graph unification is the sufficient satisfiability and combination function. Well-known applications include
3957:
3925:
3909:
3872:
3799:
3202:, in which the vertices represent the pores and the edges represent the smaller channels connecting the pores.
3166:
3044:
can be represented by a directed graph, in which the vertices represent web pages and directed edges represent
129:
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related
124:
3437:, where he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams:
7785:
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4190:
The original set cover problem, also called hitting set, can be described as a vertex cover in a hypergraph.
3913:
3847:
3573:
The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of
3390:
3251:
3134:
68:
6860:
3323:
A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or
2179:, not allowed under the definition above, are two or more edges with both the same tail and the same head.
7850:
7681:
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4500:
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543:
531:
130:
41:
32:
5426:"graphsim: An R package for simulating gene expression data from graph structures of biological pathways"
4800:
4132:
There are numerous problems arising especially from applications that have to do with various notions of
3672:, the crossing number is zero by definition. Drawings on surfaces other than the plane are also studied.
8023:
7820:
7790:
7780:
7676:
7590:
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7406:
7373:
7363:
7246:
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7007:
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6622:
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6470:
6465:
6442:
6324:
5609:
5243:
Vecchio, F (2013). "Brain network connectivity assessed using graph theory in frontotemporal dementia".
4639:
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822:
to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph)
73:
6831:
3545:
published a method for solving the problem using computers. A computer-aided proof produced in 1976 by
3470:
141:
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to the product of in- or co-variants whose separate graphs are given. " (italics as in the original).
3280:
3174:
3070:
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112:
27:
This article is about sets of vertices connected by edges. For graphs of mathematical functions, see
2592:. So to allow loops the definitions must be expanded. For directed simple graphs, the definition of
1534:
940:. To allow loops, the definitions must be expanded. For undirected simple graphs, the definition of
111:, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in
7845:
7770:
7686:
7671:
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7216:
7173:
7168:
7065:
7055:
7027:
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6720:
6642:
6541:
6475:
6432:
6422:
6402:
4617:
4602:
4572:
4450:
4252:
3860:
3811:
3737:, a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the
3513:
3403:
3190:
2445:
2189:
825:
798:
568:
28:
6219:
6152:
6125:
5916:
4715:
3315:. Algebraic graph theory has been applied to many areas including dynamic systems and complexity.
2469:
to itself is the edge (for a directed simple graph) or is incident on (for a directed multigraph)
7810:
7709:
7585:
7542:
7451:
7393:
7378:
7368:
7153:
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6755:
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6556:
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5811:
Heinrich Heesch: Untersuchungen zum
Vierfarbenproblem. Mannheim: Bibliographisches Institut 1969.
5683:
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4542:
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4013:
3680:
3501:
3485:
3477:
3284:
3212:
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4815:
3617:
3185:. This approach is especially used in computer processing of molecular structures, ranging from
3620:
of the asymptotic probability of graph connectivity, gave rise to yet another branch, known as
1287:
of a vertex is the number of edges that are incident to it, where a loop is counted twice. The
7830:
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7701:
7509:
7476:
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6013:
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5404:
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5217:
5138:
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4597:
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4167:
3563:
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3126:
3114:
3106:
2292:
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671:
633:
316:
218:
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3097:
and compositional semantics follow tree-based structures, whose expressive power lies in the
1560:
1497:
349:
159:
7997:
7979:
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7547:
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3734:
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3433:
3357:
3223:
3118:
3032:
2417:{\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}
1726:{\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}
6254:
Phase
Transitions in Combinatorial Optimization Problems, Section 3: Introduction to Graphs
6028:
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2108:
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6967:
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5857:
Appel, K.; Haken, W. (1977), "Every planar map is four colorable. Part II. Reducibility",
4865:
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4420:
3762:
3582:
3542:
3521:
3505:
3481:
3228:
3207:
3186:
3170:
3058:
3020:
2834:
1352:
4855:
4284:
Ascertaining relationships among classes (e.g. does one property of graphs imply another)
3529:
3416:
1252:
1218:
6233:
5699:
5444:
5318:
5028:
4965:
3675:
There are other techniques to visualize a graph away from vertices and edges, including
1811:
71:
used to model pairwise relations between objects. A graph in this context is made up of
7840:
7744:
7643:
7489:
7461:
6783:
6710:
6417:
6076:
6006:
5758:
5528:
5495:
5092:
5059:
4820:
4805:
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4455:
4430:
4415:
4400:
4395:
4360:
4350:
4174:
4028:
4003:
3969:
3935:
3819:
3727:
3702:
3574:
3550:
3344:
3324:
3093:, since natural language often lends itself well to discrete structure. Traditionally,
3066:
2980:
2960:
2936:
2916:
2864:
2840:
2816:
2595:
2452:
2274:
2236:
2175:
2088:
2068:
2044:
2024:
1997:
1973:
1949:
1929:
1909:
1889:
1834:
1601:
1529:
1473:
1453:
1401:
1377:
1334:
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943:
805:
779:
653:
615:
511:
492:
472:
448:
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381:
332:
200:
5886:
Robertson, N.; Sanders, D.; Seymour, P.; Thomas, R. (1997), "The four color theorem",
3896:
A similar problem, the subdivision containment problem, is to find a fixed graph as a
3612:
The introduction of probabilistic methods in graph theory, especially in the study of
3449:
diagram or chemicograph. I give a rule for the geometrical multiplication of graphs,
8017:
7941:
7729:
7017:
6571:
6503:
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5763:
Fractal Music, Hypercards, and more…Mathematical
Recreations from Scientific American
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46:
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3512:
led to the study of the colorings of the graphs embedded on surfaces with arbitrary
3492:
the same year. Many incorrect proofs have been proposed, including those by Cayley,
3303:
In mathematics, graphs are useful in geometry and certain parts of topology such as
7825:
7484:
6513:
6412:
5229:
5044:
4790:
4780:
4710:
4666:
4644:
4133:
4110:
4032:
3961:
3921:
3868:
3749:
is a modified form of the adjacency matrix that incorporates information about the
3711:
3669:
3622:
3493:
3466:
3312:
3292:
2429:
1762:
154:
6201:
5177:
3335:
509:. A vertex may exist in a graph and not belong to an edge. Under this definition,
6223:
6181:
5946:
5628:
5256:
5036:
2585:{\displaystyle \left\{(x,y)\mid (x,y)\in V^{2}\;{\textrm {and}}\;x\neq y\right\}}
8002:
7815:
7441:
7353:
6715:
6379:
6302:
6207:
5723:
5158:"A social network analysis of Twitter: Mapping the digital humanities community"
5005:
Mashaghi, A.; et al. (2004). "Investigation of a protein complex network".
4860:
4671:
3658:
3528:. The works of Ramsey on colorations and more specially the results obtained by
3304:
3090:
3036:
1121:
54:
5134:
5117:
17:
7835:
7765:
7358:
7091:
6947:
6700:
6579:
6374:
5453:
5213:
4688:
4460:
4239:, a decomposition into a collection of cycles covering each edge exactly twice
4230:
4056:
Constraint modeling theories concern families of directed graphs related by a
3802:
in a given graph. One reason to be interested in such a question is that many
3255:
people work together in a particular way, such as acting in a movie together.
3089:
Graph-theoretic methods, in various forms, have proven particularly useful in
789:
6265:
6169:
5871:
5842:
5668:
5519:
5472:
5334:
5142:
5083:
4187:
is the special case of set cover problem where sets to cover are every edges.
767:{\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}
308:{\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}
7949:
7340:
7301:
5789:
4973:
4735:
4705:
4288:
3723:
3706:
3408:
3239:
3231:
3158:
3050:
3045:
5900:
5537:
5511:
5264:
5221:
5101:
5075:
4177:
problem is the special case of set cover problem where sets are the closed
7904:
6270:
5496:"Characterizing the role of the structural connectome in seizure dynamics"
5383:"Social network analysis and visualization: Moreno’s Sociograms revisited"
5060:"Characterizing the role of the structural connectome in seizure dynamics"
3541:
The four color problem remained unsolved for more than a century. In 1969
7974:
7954:
7401:
6604:
6523:
6450:
6253:
5019:
3586:
3379:
3276:
3053:
to handle graphs is therefore of major interest in computer science. The
515:, in which two or more edges connect the same vertices, are not allowed.
6259:
5463:
3664:
Graph drawing also can be said to encompass problems that deal with the
3101:, modeled in a hierarchical graph. More contemporary approaches such as
2432:
of vertices (that is, an edge is associated with two distinct vertices).
1765:
of vertices (that is, an edge is associated with two distinct vertices).
782:
of vertices (that is, an edge is associated with two distinct vertices).
522:
Example of simple undirected graph with 3 vertices, 3 edges and 4 loops.
335:
of vertices (that is, an edge is associated with two distinct vertices).
6389:
5568:
Cauchy, A. L. (1813), "Recherche sur les polyèdres - premier mémoire",
4072:
3598:
3476:
One of the most famous and stimulating problems in graph theory is the
3264:
3162:
3146:
3142:
3041:
2796:. To avoid ambiguity, these types of objects may be called precisely a
5326:
7964:
5708:
4226:
Some specific decomposition problems that have been studied include:
3585:. Another important factor of common development of graph theory and
3378:, and represents the beginning of the branch of mathematics known as
3094:
5652:
4946:
Hale, Scott A. (2014). "Multilinguals and
Knowledge (XXG) editing".
3065:
systems focusing on rule-based in-memory manipulation of graphs are
2105:. A vertex may exist in a graph and not belong to an edge. The edge
6884:
5585:
L'Huillier, S.-A.-J. (1812–1813), "Mémoire sur la polyèdrométrie",
3397:
was led by an interest in particular analytical forms arising from
786:
To avoid ambiguity, this type of object may be called precisely an
549:
For vertices U,V,W and X, the degrees are 2,2,3 and 1 respectively.
339:
To avoid ambiguity, this type of object may be called precisely an
7959:
6213:
6030:
Graph Theory with
Applications to Engineering and Computer Science
4956:
3496:, and others. The study and the generalization of this problem by
3334:
3222:
3004:
2789:{\displaystyle \phi :E\to \left\{(x,y)\mid (x,y)\in V^{2}\right\}}
2684:{\displaystyle E\subseteq \left\{(x,y)\mid (x,y)\in V^{2}\right\}}
2436:
To avoid ambiguity, this type of object may be called precisely a
1805:
1769:
To avoid ambiguity, this type of object may be called precisely a
933:{\displaystyle \{\{x,y\}\mid x,y\in V\;{\textrm {and}}\;x\neq y\}}
6197:
Concise, annotated list of graph theory resources for researchers
4929:
4927:
2182:
In one more general sense of the term allowing multiple edges, a
561:
In one more general sense of the term allowing multiple edges, a
5948:
Lists, Decisions and Graphs. With an Introduction to Probability
3178:
7908:
6888:
6274:
6210:— non-technical paper discussing graphs of people and computers
3626:, which has been a fruitful source of graph-theoretic results.
3441:" Every invariant and co-variant thus becomes expressible by a
3516:. Tait's reformulation generated a new class of problems, the
537:
For vertices A,B,C and D, the degrees are respectively 4,4,5,1
3532:
in 1941 was at the origin of another branch of graph theory,
3138:
3753:
of the vertices, and is useful in some calculations such as
3385:
More than one century after Euler's paper on the bridges of
5118:"Applications of Graph Theory [Scanning the Issue]"
1110:. To avoid ambiguity, these types of objects may be called
6243:
with references and links to graph library implementations
1291:
of a graph is the maximum of the degrees of its vertices.
5823:"Every planar map is four colorable. Part I. Discharging"
1331:
The edges of an undirected simple graph permitting loops
3484:
in 1852 and its first written record is in a letter of
1553:
In one restricted but very common sense of the term, a
149:
In one restricted but very common sense of the term, a
5385:. Redesigned network strictly based on Moreno (1934),
3133:) and morphology (e.g. finite-state morphology, using
2813:
The edges of a directed simple graph permitting loops
4948:
Proceedings of the 2014 ACM conference on Web science
2983:
2963:
2939:
2919:
2887:
2867:
2843:
2819:
2717:
2697:
2618:
2598:
2507:
2475:
2455:
2327:
2277:
2239:
2192:
2147:
2111:
2091:
2071:
2047:
2027:
2000:
1976:
1952:
1932:
1912:
1892:
1860:
1837:
1814:
1779:
1642:
1604:
1563:
1500:
1476:
1456:
1424:
1404:
1380:
1360:
1337:
1255:
1221:
1197:
1177:
1153:
1133:
1048:
1028:
966:
946:
872:
828:
808:
694:
656:
618:
571:
495:
475:
451:
431:
404:
384:
352:
241:
203:
162:
5610:"On the theory of the analytical forms called trees"
4216:
specified trees having, respectively, 1, 2, 3, ...,
3818:
Finding the largest complete subgraph is called the
7988:
7940:
7753:
7700:
7662:
7609:
7571:
7533:
7475:
7392:
7338:
7300:
7245:
7182:
7115:
7079:
7036:
7000:
6933:
6774:
6731:
6641:
6603:
6570:
6522:
6494:
6441:
6388:
6345:
3407:. This study had many implications for theoretical
1103:{\displaystyle \phi :E\to \{\{x,y\}\mid x,y\in V\}}
1015:{\displaystyle E\subseteq \{\{x,y\}\mid x,y\in V\}}
107:, where edges link two vertices symmetrically, and
6127:Graph Algorithms in The Language of Linear Algebra
6005:
5918:Graph Algorithms in the Language of Linear Algebra
5608:
5287:
5116:
3948:Another problem in subdivision containment is the
3461:The first textbook on graph theory was written by
3427:In particular, the term "graph" was introduced by
2989:
2969:
2945:
2925:
2905:
2873:
2849:
2825:
2788:
2703:
2683:
2604:
2584:
2493:
2461:
2416:
2283:
2245:
2222:
2165:
2129:
2097:
2077:
2053:
2033:
2006:
1982:
1958:
1938:
1918:
1898:
1878:
1843:
1823:
1792:
1725:
1610:
1587:
1512:
1482:
1462:
1442:
1410:
1386:
1366:
1343:
1271:
1237:
1203:
1183:
1159:
1139:
1102:
1034:
1014:
952:
932:
858:
814:
766:
662:
624:
601:
501:
481:
457:
437:
410:
390:
370:
307:
209:
186:
4233:, a decomposition into as few forests as possible
3842:Finding the largest edgeless induced subgraph or
3705:used depends on both the graph structure and the
3419:between 1935 and 1937. These were generalized by
3283:. Another use is to model genes or proteins in a
4933:
4909:
4884:
4613:Tarjan's strongly connected components algorithm
3826:One special case of subgraph isomorphism is the
3157:Graph theory is also used to study molecules in
1022:. For undirected multigraphs, the definition of
6182:A searchable database of small connected graphs
5945:Bender, Edward A.; Williamson, S. Gill (2010).
5777:Society for Industrial and Applied Mathematics
5657:Berichte der Deutschen Chemischen Gesellschaft
2691:. For directed multigraphs, the definition of
7920:
6924:Note: This template roughly follows the 2012
6900:
6286:
4063:For constraint frameworks which are strictly
3141:, as well as various 'Net' projects, such as
1550:is a graph in which edges have orientations.
8:
6262:2007 by Jorgen Bang-Jensen and Gregory Gutin
6260:Digraphs: Theory Algorithms and Applications
3924:if it contains as a subdivision neither the
3411:. The techniques he used mainly concern the
1851:elements that are not necessarily distinct.
1097:
1076:
1064:
1061:
1009:
988:
976:
973:
927:
888:
876:
873:
853:
847:
841:
829:
761:
722:
710:
707:
365:
353:
302:
263:
251:
248:
145:A graph with three vertices and three edges.
6073:Algorithmic Graph Theory and Perfect Graphs
5782:Looking Back, Looking Ahead: A SIAM History
4097:(also called the "Chinese postman problem")
3401:to study a particular class of graphs, the
3105:model the syntax of natural language using
555:Examples of finding the degree of vertices.
7927:
7913:
7905:
6907:
6893:
6885:
6293:
6279:
6271:
5732:, Cambridge University Press, p. 30,
5115:Adali, Tulay; Ortega, Antonio (May 2018).
3714:as they have smaller memory requirements.
3268:to the movement can affect other species.
3125:. Still, other methods in phonology (e.g.
2567:
2559:
2399:
2391:
1708:
1700:
917:
909:
751:
743:
292:
284:
6102:Mahadev, N. V. R.; Peled, Uri N. (1995).
6088:. Reading, Massachusetts: Addison-Wesley.
5967:Biggs, N.; Lloyd, E.; Wilson, R. (1986).
5899:
5888:Journal of Combinatorial Theory, Series B
5870:
5841:
5707:
5551:Biggs, N.; Lloyd, E.; Wilson, R. (1986),
5527:
5462:
5452:
5401:Discrete mathematics and its applications
5357:. Oxford University Press. Archived from
5176:
5091:
5018:
4955:
4329:List of unsolved problems in graph theory
3726:, an array of pairs of vertices, and the
3465:, and published in 1936. Another book by
3393:was introducing the concept of topology,
2982:
2962:
2938:
2918:
2886:
2866:
2842:
2818:
2775:
2716:
2696:
2670:
2617:
2597:
2561:
2560:
2553:
2506:
2474:
2454:
2393:
2392:
2385:
2326:
2276:
2238:
2191:
2146:
2110:
2090:
2070:
2046:
2026:
1999:
1975:
1951:
1931:
1911:
1891:
1859:
1836:
1813:
1784:
1778:
1702:
1701:
1694:
1641:
1603:
1562:
1499:
1475:
1455:
1423:
1403:
1379:
1359:
1336:
1264:
1256:
1254:
1230:
1222:
1220:
1196:
1176:
1152:
1132:
1047:
1027:
965:
945:
911:
910:
871:
827:
807:
745:
744:
693:
655:
617:
570:
494:
474:
450:
430:
403:
383:
351:
286:
285:
240:
202:
161:
6093:Harary, Frank; Palmer, Edgar M. (1973).
6036:. Englewood, New Jersey: Prentice-Hall.
5765:, W. H. Freeman and Company, p. 203
5424:Kelly, S.; Black, Michael (2020-07-09).
3790:Subgraphs, induced subgraphs, and minors
1533:
1112:undirected simple graph permitting loops
517:
140:
40:
6216:— tools to teach and learn graph theory
5995:Bollobás, Béla; Riordan, O. M. (2003).
5958:Théorie des graphes et ses applications
5403:(7th ed.). New York: McGraw-Hill.
4921:See, for instance, Graham et al., p. 5.
4896:See, for instance, Iyanaga and Kawada,
4877:
4259:into regular subgraphs of given degrees
3171:Feynman graphs and rules of calculation
3057:is often formalized and represented by
1298:, the maximum degree of each vertex is
1294:In an undirected simple graph of order
49:of a graph with 6 vertices and 7 edges.
7624:Knowledge representation and reasoning
6124:Kepner, Jeremy; Gilbert, John (2011).
5976:Bondy, J. A.; Murty, U. S. R. (2008).
5915:Kepner, Jeremy; Gilbert, John (2011).
5294:. New York: McGraw-Hill. p. viii.
3871:if it contains as a minor neither the
3745:indicates the degree of vertices. The
2798:directed simple graph permitting loops
1191:is often assumed to be non-empty, but
1116:undirected multigraph permitting loops
7649:Philosophy of artificial intelligence
6097:. New York, New York: Academic Press.
5286:Bjorken, J. D.; Drell, S. D. (1965).
1305:and the maximum size of the graph is
7:
6968:Energy consumption (Green computing)
6855:
6130:. Philadelphia, Pennsylvania: SIAM.
5999:(1st ed.). Weinheim: Wiley VCH.
3263:Likewise, graph theory is useful in
3238:Graph theory is also widely used in
3103:head-driven phrase structure grammar
2802:directed multigraph permitting loops
1211:is allowed to be the empty set. The
7654:Distributed artificial intelligence
6926:ACM Computing Classification System
6867:
6104:Threshold Graphs and Related Topics
4608:Push–relabel maximum flow algorithm
4278:Characterizing a class in terms of
4073:elaboration of linguistic structure
7159:Integrated development environment
5779:(2002), "The George Polya Prize",
4170:on subsets of vertices/subgraphs.
4045:Hadwiger conjecture (graph theory)
3683:, and other visualizations of the
3281:single-cell transcriptome analysis
1773:. In set theory and graph theory,
25:
7634:Automated planning and scheduling
7164:Software configuration management
6266:Graph Theory, by Reinhard Diestel
3271:Graphs are also commonly used in
2957:to one another, which is denoted
2018:of the edge. The edge is said to
1494:to one another, which is denoted
422:of the edge. The edge is said to
103:). A distinction is made between
7970:Microsoft Automatic Graph Layout
7888:
7878:
7869:
7868:
6866:
6854:
6843:
6842:
6830:
6193: (archived February 6, 2006)
6053:from the original on 2019-05-17.
5682:Sylvester, James Joseph (1878).
5571:Journal de l'École Polytechnique
5399:Rosen, Kenneth H. (2011-06-14).
3798:, is finding a fixed graph as a
3431:in a paper published in 1878 in
542:
530:
7879:
7282:Computational complexity theory
6751:Computational complexity theory
5433:Journal of Open Source Software
4166:in graphs may refer to various
3782:There is a large literature on
7066:Network performance evaluation
5439:(51). The Open Journal: 2161.
4245:, a decomposition into as few
3733:Matrix structures include the
3691:Tabular: Graph data structures
2900:
2888:
2881:. Specifically, for each edge
2765:
2753:
2747:
2735:
2727:
2660:
2648:
2642:
2630:
2543:
2531:
2525:
2513:
2488:
2476:
2375:
2363:
2357:
2345:
2337:
2217:
2199:
2160:
2148:
2124:
2112:
1873:
1861:
1831:that is, ordered sequences of
1684:
1672:
1666:
1654:
1582:
1570:
1437:
1425:
1418:. Specifically, for each edge
1265:
1257:
1245:, its number of vertices. The
1231:
1223:
1058:
704:
596:
578:
181:
169:
1:
7437:Multimedia information system
7422:Geographic information system
7412:Enterprise information system
7001:Computer systems organization
5821:Appel, K.; Haken, W. (1977),
5178:10.1080/23311983.2016.1171458
4024:Erdős–Faber–Lovász conjecture
3833:A similar problem is finding
3794:A common problem, called the
3339:The Königsberg Bridge problem
3291:Graph theory is also used in
3250:, notably through the use of
3099:principle of compositionality
2223:{\displaystyle G=(V,E,\phi )}
859:{\displaystyle \{x,x\}=\{x\}}
602:{\displaystyle G=(V,E,\phi )}
7796:Computational social science
7384:Theoretical computer science
7197:Software development process
6973:Electronic design automation
6958:Very Large Scale Integration
6256:(2006) by Hartmann and Weigt
5788:, p. 26, archived from
5629:10.1017/CBO9780511703690.046
5257:10.1212/WNL.0b013e31829a33f8
5165:Cogent Arts & Humanities
4934:Bender & Williamson 2010
4910:Bender & Williamson 2010
4885:Bender & Williamson 2010
4656:Related areas of mathematics
4598:Planarity testing algorithms
4334:Publications in graph theory
4019:Strong perfect graph theorem
3796:subgraph isomorphism problem
3722:List structures include the
3593:, who published in 1845 his
38:Area of discrete mathematics
7619:Natural language processing
7407:Information storage systems
6158:Encyclopedia of Mathematics
5290:Relativistic Quantum Fields
5007:European Physical Journal B
4694:Abstract simplicial complex
4588:Nearest neighbour algorithm
4366:Disjoint-set data structure
4324:List of graph theory topics
4101:Seven bridges of Königsberg
4052:Subsumption and unification
3445:precisely identical with a
3349:Seven Bridges of Königsberg
3227:Graph theory in sociology:
3121:are therefore important in
1279:, its number of edges. The
8040:
7535:Human–computer interaction
7505:Intrusion detection system
7417:Social information systems
7402:Database management system
6801:Films about mathematicians
6234:A list of graph algorithms
6208:The Social Life of Routers
6119:. Oxford University Press.
6064:Cambridge University Press
5971:. Oxford University Press.
5381:Grandjean, Martin (2015).
5307:Journal of Applied Physics
5202:Brain Imaging and Behavior
5156:Grandjean, Martin (2016).
5135:10.1109/JPROC.2018.2820300
5037:10.1140/epjb/e2004-00301-0
4645:Probabilistic graph theory
4121:Traveling salesman problem
4001:
3950:Kelmans–Seymour conjecture
3697:Graph (abstract data type)
3694:
3642:
3520:, particularly studied by
3242:as a way, for example, to
3195:computational neuroscience
3189:to database searching. In
1527:
122:
26:
7864:
7801:Computational engineering
7776:Computational mathematics
6922:
6824:
6370:Philosophy of mathematics
6310:
6117:Networks: An Introduction
6071:Golumbic, Martin (1980).
6008:Introductory Graph Theory
5921:. SIAM. p. 1171458.
5557:, Oxford University Press
5454:10.1101/2020.03.02.972471
5351:Networks: An Introduction
5214:10.1007/s11682-016-9528-3
4700:Prominent graph theorists
4593:Network simplex algorithm
4069:automatic theorem proving
4029:Total coloring conjecture
3992:reconstruction conjecture
3885:) nor the complete graph
3828:graph isomorphism problem
3123:computational linguistics
2428:mapping every edge to an
778:mapping every edge to an
85:) which are connected by
7811:Computational healthcare
7806:Differentiable computing
7725:Graphics processing unit
7144:Domain-specific language
7013:Computational complexity
6806:Recreational mathematics
6222:, and library resources
6060:Algorithmic Graph Theory
6004:Chartrand, Gary (1985).
5669:10.1002/cber.18750080252
5587:Annales de Mathématiques
4900:, p. 234 or Biggs, p. 4.
4650:Topological graph theory
4563:Ford–Fulkerson algorithm
4558:Floyd–Warshall algorithm
4319:Glossary of graph theory
4141:Max flow min cut theorem
4095:Route inspection problem
4085:Hamiltonian path problem
4039:List coloring conjecture
4035:'s conjecture (unsolved)
3926:complete bipartite graph
3873:complete bipartite graph
3595:Kirchhoff's circuit laws
3244:measure actors' prestige
3167:condensed matter physics
3135:finite-state transducers
3107:typed feature structures
3077:storing and querying of
3055:transformation of graphs
125:Glossary of graph theory
7935:Graph analysis software
7786:Computational chemistry
7720:Photograph manipulation
7611:Artificial intelligence
7427:Decision support system
6691:Mathematical statistics
6681:Mathematical psychology
6651:Engineering mathematics
6585:Algebraic number theory
5969:Graph Theory, 1736–1936
5684:"Chemistry and Algebra"
5574:, 9 (Cahier 16): 66–86.
5554:Graph Theory, 1736-1936
5123:Proceedings of the IEEE
4974:10.1145/2615569.2615684
4568:Hopcroft–Karp algorithm
4501:Strongly regular graphs
4314:Gallery of named graphs
4280:forbidden substructures
4255:, a decomposition of a
3984:point-deleted subgraphs
3848:independent set problem
3657:The pioneering work of
3252:social network analysis
3111:directed acyclic graphs
1588:{\displaystyle G=(V,E)}
1513:{\displaystyle x\sim y}
371:{\displaystyle \{x,y\}}
341:undirected simple graph
187:{\displaystyle G=(V,E)}
131:mathematical structures
69:mathematical structures
7851:Educational technology
7682:Reinforcement learning
7432:Process control system
7330:Computational geometry
7320:Algorithmic efficiency
7315:Analysis of algorithms
6963:Systems on Chip (SoCs)
6837:Mathematics portal
6686:Mathematical sociology
6666:Mathematical economics
6661:Mathematical chemistry
6590:Analytic number theory
6471:Differential equations
6084:Harary, Frank (1969).
6058:Gibbons, Alan (1985).
6027:Deo, Narsingh (1974).
5956:Berge, Claude (1958).
5901:10.1006/jctb.1997.1750
5872:10.1215/ijm/1256049012
5843:10.1215/ijm/1256049011
5616:Philosophical Magazine
4635:Geometric graph theory
4630:Algebraic graph theory
4553:Edmonds–Karp algorithm
4528:Bellman–Ford algorithm
4466:Pebble motion problems
4426:Graph sandwich problem
4346:Algebraic graph theory
4302:for members of a class
4275:the members of a class
4195:Decomposition problems
3769:between two vertices.
3518:factorization problems
3340:
3309:Algebraic graph theory
3235:
3010:
2991:
2971:
2947:
2927:
2907:
2875:
2851:
2827:
2790:
2711:should be modified to
2705:
2685:
2612:should be modified to
2606:
2586:
2495:
2463:
2418:
2285:
2247:
2224:
2167:
2131:
2099:
2079:
2055:
2035:
2008:
1984:
1960:
1940:
1920:
1900:
1880:
1845:
1825:
1794:
1727:
1612:
1589:
1539:
1514:
1484:
1464:
1444:
1412:
1388:
1368:
1345:
1273:
1239:
1205:
1185:
1161:
1141:
1104:
1042:should be modified to
1036:
1016:
960:should be modified to
954:
934:
860:
816:
768:
664:
626:
603:
523:
503:
483:
459:
439:
412:
392:
372:
309:
211:
188:
146:
50:
33:Graph (disambiguation)
31:. For other uses, see
7821:Electronic publishing
7791:Computational biology
7781:Computational physics
7677:Unsupervised learning
7591:Distributed computing
7467:Information retrieval
7374:Mathematical analysis
7364:Mathematical software
7247:Theory of computation
7212:Software construction
7202:Requirements analysis
7080:Software organization
7008:Computer architecture
6978:Hardware acceleration
6943:Printed circuit board
6816:Mathematics education
6746:Theory of computation
6466:Hypercomplex analysis
6214:Graph Theory Software
6187:Image gallery: graphs
6170:Graph theory tutorial
6115:Newman, Mark (2010).
6095:Graphical Enumeration
5348:Newman, Mark (2010).
4741:Dirac, Gabriel Andrew
4640:Extremal graph theory
4496:Spectral graph theory
4486:Random regular graphs
4295:membership in a class
4116:Three-cottage problem
4106:Shortest path problem
4090:Minimum spanning tree
3883:Three-cottage problem
3784:graphical enumeration
3639:Visual: Graph drawing
3535:extremal graph theory
3413:enumeration of graphs
3399:differential calculus
3343:The paper written by
3338:
3311:has close links with
3226:
3204:Chemical graph theory
3153:Physics and chemistry
3079:graph-structured data
3059:graph rewrite systems
3008:
2992:
2972:
2948:
2928:
2908:
2906:{\displaystyle (x,y)}
2876:
2852:
2837:~ on the vertices of
2828:
2791:
2706:
2704:{\displaystyle \phi }
2686:
2607:
2587:
2496:
2494:{\displaystyle (x,x)}
2464:
2419:
2286:
2248:
2225:
2186:is an ordered triple
2168:
2166:{\displaystyle (x,y)}
2132:
2130:{\displaystyle (y,x)}
2100:
2080:
2056:
2036:
2009:
1985:
1961:
1941:
1921:
1901:
1881:
1879:{\displaystyle (x,y)}
1846:
1826:
1795:
1793:{\displaystyle V^{n}}
1771:directed simple graph
1728:
1613:
1590:
1537:
1515:
1485:
1465:
1445:
1443:{\displaystyle (x,y)}
1413:
1389:
1369:
1367:{\displaystyle \sim }
1346:
1274:
1240:
1206:
1186:
1162:
1142:
1105:
1037:
1035:{\displaystyle \phi }
1017:
955:
935:
861:
817:
769:
665:
627:
604:
565:is an ordered triple
521:
504:
484:
460:
440:
413:
393:
373:
310:
212:
189:
144:
123:Further information:
44:
7581:Concurrent computing
7553:Ubiquitous computing
7525:Application security
7520:Information security
7349:Discrete mathematics
7325:Randomized algorithm
7277:Computability theory
7255:Model of computation
7227:Software maintenance
7222:Software engineering
7184:Software development
7134:Programming language
7129:Programming paradigm
7046:Network architecture
6796:Informal mathematics
6676:Mathematical physics
6671:Mathematical finance
6656:Mathematical biology
6595:Diophantine geometry
6204:— a graph theory IDE
5512:10.1093/brain/awz125
5076:10.1093/brain/awz125
4578:Kosaraju's algorithm
4548:Dijkstra's algorithm
4538:Breadth-first search
4511:Transitive reduction
4406:Graph data structure
4371:Dual-phase evolution
4185:Vertex cover problem
4153:Museum guard problem
3914:Kuratowski's Theorem
3900:of a given graph. A
3597:for calculating the
3362:carried on with the
3175:quantum field theory
3063:graph transformation
2981:
2961:
2937:
2917:
2885:
2865:
2841:
2835:homogeneous relation
2817:
2715:
2695:
2616:
2596:
2505:
2473:
2453:
2325:
2275:
2237:
2190:
2145:
2109:
2089:
2069:
2045:
2025:
1998:
1974:
1950:
1930:
1910:
1890:
1858:
1835:
1812:
1777:
1640:
1602:
1561:
1498:
1474:
1454:
1422:
1402:
1378:
1358:
1353:homogeneous relation
1335:
1253:
1219:
1195:
1175:
1151:
1131:
1046:
1026:
964:
944:
870:
826:
806:
692:
654:
616:
569:
493:
473:
449:
429:
402:
382:
350:
239:
201:
160:
113:discrete mathematics
7856:Document management
7846:Operations research
7771:Enterprise software
7687:Multi-task learning
7672:Supervised learning
7394:Information systems
7217:Software deployment
7174:Software repository
7028:Real-time computing
6811:Mathematics and art
6721:Operations research
6476:Functional analysis
5700:1878Natur..17..284S
5651:Cayley, A. (1875),
5464:10.21105/joss.02161
5445:2020JOSS....5.2161K
5319:2016JAP...119a5102K
5029:2004EPJB...41..113M
4966:2013arXiv1312.0976H
4950:. pp. 99–108.
4786:Heawood, Percy John
4766:Fleischner, Herbert
4746:Dijkstra, Edsger W.
4618:Topological sorting
4583:Kruskal's algorithm
4573:Hungarian algorithm
4533:Borůvka's algorithm
4516:Tree data structure
4253:Graph factorization
4147:Visibility problems
3812:NP-complete problem
3755:Kirchhoff's theorem
3623:random graph theory
3453:for constructing a
3191:statistical physics
3061:. Complementary to
2857:that is called the
2438:directed multigraph
1800:denotes the set of
1557:is an ordered pair
1394:that is called the
1374:on the vertices of
1351:induce a symmetric
1272:{\displaystyle |E|}
1238:{\displaystyle |V|}
29:Graph of a function
7639:Search methodology
7586:Parallel computing
7543:Interaction design
7452:Computing platform
7379:Numerical analysis
7369:Information theory
7154:Software framework
7117:Software notations
7056:Network components
6953:Integrated circuit
6756:Numerical analysis
6365:Mathematical logic
6360:Information theory
6239:2019-07-13 at the
6230:about graph theory
6228:in other libraries
6175:2012-01-16 at the
4851:Thomassen, Carsten
4811:Nešetřil, Jaroslav
4726:Brightwell, Graham
4721:Bondy, Adrian John
4543:Depth-first search
4471:Percolation theory
4436:Intersection graph
4391:Graph automorphism
4287:Finding efficient
4237:Cycle double cover
4168:set cover problems
4014:Four-color theorem
3960:graph that is not
3958:5-vertex-connected
3681:intersection graph
3478:four color problem
3341:
3236:
3213:percolation theory
3011:
2987:
2967:
2943:
2923:
2903:
2871:
2859:adjacency relation
2847:
2823:
2786:
2701:
2681:
2602:
2582:
2491:
2459:
2426:incidence function
2414:
2281:
2243:
2220:
2163:
2127:
2095:
2075:
2051:
2031:
2004:
1980:
1956:
1936:
1916:
1896:
1876:
1841:
1824:{\displaystyle V,}
1821:
1790:
1723:
1608:
1585:
1540:
1510:
1480:
1460:
1440:
1408:
1396:adjacency relation
1384:
1364:
1341:
1269:
1235:
1201:
1181:
1157:
1137:
1100:
1032:
1012:
950:
930:
856:
812:
776:incidence function
764:
660:
622:
599:
524:
499:
479:
455:
435:
408:
388:
368:
305:
207:
184:
147:
51:
8011:
8010:
7902:
7901:
7831:Electronic voting
7761:Quantum Computing
7754:Applied computing
7740:Image compression
7510:Hardware security
7500:Security services
7457:Digital marketing
7237:Open-source model
7149:Modeling language
7061:Network scheduler
6882:
6881:
6481:Harmonic analysis
6137:978-0-898719-90-1
5987:978-1-84628-969-9
5928:978-0-898719-90-1
5859:Illinois J. Math.
5830:Illinois J. Math.
5739:978-0-521-79489-3
5410:978-0-07-338309-5
5387:Who Shall Survive
5327:10.1063/1.4939280
4731:Chudnovsky, Maria
4491:Semantic networks
4381:Existential graph
4164:Covering problems
4159:Covering problems
4134:flows in networks
4071:and modeling the
3835:induced subgraphs
3757:on the number of
3607:electric circuits
3273:molecular biology
3127:optimality theory
3119:semantic networks
3115:lexical semantics
2990:{\displaystyle y}
2970:{\displaystyle x}
2946:{\displaystyle y}
2926:{\displaystyle x}
2874:{\displaystyle G}
2850:{\displaystyle G}
2826:{\displaystyle G}
2605:{\displaystyle E}
2564:
2462:{\displaystyle x}
2396:
2284:{\displaystyle E}
2246:{\displaystyle V}
2098:{\displaystyle y}
2078:{\displaystyle x}
2054:{\displaystyle y}
2034:{\displaystyle x}
2007:{\displaystyle y}
1983:{\displaystyle x}
1959:{\displaystyle y}
1939:{\displaystyle x}
1919:{\displaystyle y}
1899:{\displaystyle x}
1844:{\displaystyle n}
1705:
1611:{\displaystyle V}
1483:{\displaystyle y}
1463:{\displaystyle x}
1411:{\displaystyle G}
1387:{\displaystyle G}
1344:{\displaystyle G}
1204:{\displaystyle E}
1184:{\displaystyle V}
1160:{\displaystyle E}
1140:{\displaystyle V}
1125:), respectively.
953:{\displaystyle E}
914:
815:{\displaystyle x}
748:
663:{\displaystyle E}
625:{\displaystyle V}
502:{\displaystyle y}
482:{\displaystyle x}
458:{\displaystyle y}
438:{\displaystyle x}
411:{\displaystyle y}
391:{\displaystyle x}
289:
210:{\displaystyle V}
105:undirected graphs
16:(Redirected from
8031:
7929:
7922:
7915:
7906:
7892:
7891:
7882:
7881:
7872:
7871:
7692:Cross-validation
7664:Machine learning
7548:Social computing
7515:Network security
7310:Algorithm design
7232:Programming team
7192:Control variable
7169:Software library
7107:Software quality
7102:Operating system
7051:Network protocol
6916:Computer science
6909:
6902:
6895:
6886:
6870:
6869:
6858:
6857:
6846:
6845:
6835:
6834:
6766:Computer algebra
6741:Computer science
6461:Complex analysis
6295:
6288:
6281:
6272:
6248:Online textbooks
6166:
6141:
6120:
6111:
6098:
6089:
6080:
6067:
6054:
6052:
6035:
6023:
6011:
6000:
5991:
5972:
5961:
5952:
5933:
5932:
5912:
5906:
5905:
5903:
5883:
5877:
5876:
5874:
5854:
5848:
5847:
5845:
5827:
5818:
5812:
5809:
5803:
5802:
5801:
5800:
5794:
5787:
5773:
5767:
5766:
5755:
5749:
5748:
5747:
5746:
5720:
5714:
5713:
5711:
5709:10.1038/017284a0
5679:
5673:
5672:
5663:(2): 1056–1059,
5648:
5642:
5641:
5612:
5601:
5595:
5594:
5582:
5576:
5575:
5565:
5559:
5558:
5548:
5542:
5541:
5531:
5506:(7): 1955–1972.
5491:
5485:
5484:
5466:
5456:
5430:
5421:
5415:
5414:
5396:
5390:
5379:
5373:
5372:
5370:
5369:
5363:
5356:
5345:
5339:
5338:
5302:
5296:
5295:
5293:
5283:
5277:
5276:
5240:
5234:
5233:
5197:
5191:
5190:
5180:
5162:
5153:
5147:
5146:
5120:
5112:
5106:
5105:
5095:
5070:(7): 1955–1972.
5055:
5049:
5048:
5022:
5020:cond-mat/0304207
5002:
4996:
4995:
4959:
4943:
4937:
4931:
4922:
4919:
4913:
4907:
4901:
4894:
4888:
4882:
4866:Whitney, Hassler
4841:Szemerédi, Endre
4771:Golumbic, Martin
4603:Prim's algorithm
4506:Symmetric graphs
4376:Entitative graph
4356:Conceptual graph
4222:
4215:
3968:of the 5-vertex
3861:Wagner's Theorem
3804:graph properties
3761:of a graph. The
3747:Laplacian matrix
3739:adjacency matrix
3735:incidence matrix
3685:adjacency matrix
3591:Gustav Kirchhoff
3187:chemical editors
3033:computer science
3027:Computer science
2996:
2994:
2993:
2988:
2976:
2974:
2973:
2968:
2952:
2950:
2949:
2944:
2932:
2930:
2929:
2924:
2913:, its endpoints
2912:
2910:
2909:
2904:
2880:
2878:
2877:
2872:
2856:
2854:
2853:
2848:
2832:
2830:
2829:
2824:
2810:) respectively.
2795:
2793:
2792:
2787:
2785:
2781:
2780:
2779:
2710:
2708:
2707:
2702:
2690:
2688:
2687:
2682:
2680:
2676:
2675:
2674:
2611:
2609:
2608:
2603:
2591:
2589:
2588:
2583:
2581:
2577:
2566:
2565:
2562:
2558:
2557:
2501:which is not in
2500:
2498:
2497:
2492:
2468:
2466:
2465:
2460:
2423:
2421:
2420:
2415:
2413:
2409:
2398:
2397:
2394:
2390:
2389:
2290:
2288:
2287:
2282:
2252:
2250:
2249:
2244:
2229:
2227:
2226:
2221:
2172:
2170:
2169:
2164:
2136:
2134:
2133:
2128:
2104:
2102:
2101:
2096:
2084:
2082:
2081:
2076:
2060:
2058:
2057:
2052:
2040:
2038:
2037:
2032:
2013:
2011:
2010:
2005:
1994:of the edge and
1989:
1987:
1986:
1981:
1965:
1963:
1962:
1957:
1945:
1943:
1942:
1937:
1925:
1923:
1922:
1917:
1905:
1903:
1902:
1897:
1885:
1883:
1882:
1877:
1850:
1848:
1847:
1842:
1830:
1828:
1827:
1822:
1803:
1799:
1797:
1796:
1791:
1789:
1788:
1732:
1730:
1729:
1724:
1722:
1718:
1707:
1706:
1703:
1699:
1698:
1617:
1615:
1614:
1609:
1594:
1592:
1591:
1586:
1519:
1517:
1516:
1511:
1489:
1487:
1486:
1481:
1469:
1467:
1466:
1461:
1450:, its endpoints
1449:
1447:
1446:
1441:
1417:
1415:
1414:
1409:
1393:
1391:
1390:
1385:
1373:
1371:
1370:
1365:
1350:
1348:
1347:
1342:
1327:
1325:
1324:
1321:
1318:
1304:
1278:
1276:
1275:
1270:
1268:
1260:
1244:
1242:
1241:
1236:
1234:
1226:
1210:
1208:
1207:
1202:
1190:
1188:
1187:
1182:
1166:
1164:
1163:
1158:
1146:
1144:
1143:
1138:
1118:(sometimes also
1109:
1107:
1106:
1101:
1041:
1039:
1038:
1033:
1021:
1019:
1018:
1013:
959:
957:
956:
951:
939:
937:
936:
931:
916:
915:
912:
866:which is not in
865:
863:
862:
857:
821:
819:
818:
813:
773:
771:
770:
765:
750:
749:
746:
669:
667:
666:
661:
631:
629:
628:
623:
608:
606:
605:
600:
546:
534:
508:
506:
505:
500:
488:
486:
485:
480:
464:
462:
461:
456:
444:
442:
441:
436:
417:
415:
414:
409:
397:
395:
394:
389:
377:
375:
374:
369:
314:
312:
311:
306:
291:
290:
287:
216:
214:
213:
208:
193:
191:
190:
185:
61:is the study of
21:
8039:
8038:
8034:
8033:
8032:
8030:
8029:
8028:
8014:
8013:
8012:
8007:
7984:
7936:
7933:
7903:
7898:
7889:
7860:
7841:Word processing
7749:
7735:Virtual reality
7696:
7658:
7629:Computer vision
7605:
7601:Multiprocessing
7567:
7529:
7495:Security hacker
7471:
7447:Digital library
7388:
7339:Mathematics of
7334:
7296:
7272:Automata theory
7267:Formal language
7241:
7207:Software design
7178:
7111:
7097:Virtual machine
7075:
7071:Network service
7032:
7023:Embedded system
6996:
6929:
6918:
6913:
6883:
6878:
6829:
6820:
6770:
6727:
6706:Systems science
6637:
6633:Homotopy theory
6599:
6566:
6518:
6490:
6437:
6384:
6355:Category theory
6341:
6306:
6299:
6250:
6241:Wayback Machine
6224:in your library
6191:Wayback Machine
6177:Wayback Machine
6151:
6148:
6138:
6123:
6114:
6101:
6092:
6083:
6070:
6057:
6050:
6044:
6033:
6026:
6020:
6003:
5994:
5988:
5975:
5966:
5960:. Paris: Dunod.
5955:
5944:
5941:
5936:
5929:
5914:
5913:
5909:
5885:
5884:
5880:
5856:
5855:
5851:
5825:
5820:
5819:
5815:
5810:
5806:
5798:
5796:
5792:
5785:
5775:
5774:
5770:
5759:Gardner, Martin
5757:
5756:
5752:
5744:
5742:
5740:
5722:
5721:
5717:
5681:
5680:
5676:
5650:
5649:
5645:
5639:
5623:(85): 172–176,
5603:
5602:
5598:
5584:
5583:
5579:
5567:
5566:
5562:
5550:
5549:
5545:
5493:
5492:
5488:
5428:
5423:
5422:
5418:
5411:
5398:
5397:
5393:
5380:
5376:
5367:
5365:
5361:
5354:
5347:
5346:
5342:
5304:
5303:
5299:
5285:
5284:
5280:
5242:
5241:
5237:
5199:
5198:
5194:
5160:
5155:
5154:
5150:
5114:
5113:
5109:
5057:
5056:
5052:
5004:
5003:
4999:
4984:
4945:
4944:
4940:
4932:
4925:
4920:
4916:
4908:
4904:
4895:
4891:
4883:
4879:
4875:
4870:
4826:Robertson, Neil
4821:Ringel, Gerhard
4806:Murty, U. S. R.
4756:Euler, Leonhard
4702:
4685:
4683:Generalizations
4658:
4626:
4524:
4421:Graph rewriting
4342:
4309:
4300:representations
4266:
4217:
4210:
4208:
4197:
4161:
4149:
4136:, for example:
4130:
4081:
4054:
4006:
4000:
3986:. For example:
3977:
3943:
3933:
3912:. For example,
3891:
3880:
3844:independent set
3814:. For example:
3792:
3780:
3775:
3763:distance matrix
3699:
3693:
3677:circle packings
3666:crossing number
3647:
3641:
3632:
3543:Heinrich Heesch
3482:Francis Guthrie
3333:
3325:weighted graphs
3321:
3301:
3261:
3248:rumor spreading
3221:
3219:Social sciences
3208:molecular graph
3155:
3087:
3069:geared towards
3067:graph databases
3029:
3021:network science
3003:
2979:
2978:
2959:
2958:
2953:are said to be
2935:
2934:
2915:
2914:
2883:
2882:
2863:
2862:
2839:
2838:
2815:
2814:
2771:
2734:
2730:
2713:
2712:
2693:
2692:
2666:
2629:
2625:
2614:
2613:
2594:
2593:
2549:
2512:
2508:
2503:
2502:
2471:
2470:
2451:
2450:
2381:
2344:
2340:
2323:
2322:
2273:
2272:
2235:
2234:
2188:
2187:
2143:
2142:
2107:
2106:
2087:
2086:
2067:
2066:
2043:
2042:
2023:
2022:
1996:
1995:
1972:
1971:
1966:are called the
1948:
1947:
1928:
1927:
1926:, the vertices
1908:
1907:
1888:
1887:
1856:
1855:
1833:
1832:
1810:
1809:
1808:of elements of
1801:
1780:
1775:
1774:
1690:
1653:
1649:
1638:
1637:
1600:
1599:
1559:
1558:
1532:
1526:
1496:
1495:
1490:are said to be
1472:
1471:
1452:
1451:
1420:
1419:
1400:
1399:
1376:
1375:
1356:
1355:
1333:
1332:
1322:
1319:
1309:
1308:
1306:
1299:
1251:
1250:
1217:
1216:
1193:
1192:
1173:
1172:
1149:
1148:
1129:
1128:
1044:
1043:
1024:
1023:
962:
961:
942:
941:
868:
867:
824:
823:
804:
803:
690:
689:
652:
651:
614:
613:
567:
566:
559:
558:
557:
556:
552:
551:
550:
547:
539:
538:
535:
491:
490:
471:
470:
447:
446:
427:
426:
418:are called the
400:
399:
380:
379:
378:, the vertices
348:
347:
333:unordered pairs
237:
236:
199:
198:
158:
157:
139:
127:
121:
109:directed graphs
39:
36:
23:
22:
18:Graph-theoretic
15:
12:
11:
5:
8037:
8035:
8027:
8026:
8016:
8015:
8009:
8008:
8006:
8005:
8000:
7994:
7992:
7986:
7985:
7983:
7982:
7977:
7972:
7967:
7962:
7957:
7952:
7946:
7944:
7938:
7937:
7934:
7932:
7931:
7924:
7917:
7909:
7900:
7899:
7897:
7896:
7886:
7876:
7865:
7862:
7861:
7859:
7858:
7853:
7848:
7843:
7838:
7833:
7828:
7823:
7818:
7813:
7808:
7803:
7798:
7793:
7788:
7783:
7778:
7773:
7768:
7763:
7757:
7755:
7751:
7750:
7748:
7747:
7745:Solid modeling
7742:
7737:
7732:
7727:
7722:
7717:
7712:
7706:
7704:
7698:
7697:
7695:
7694:
7689:
7684:
7679:
7674:
7668:
7666:
7660:
7659:
7657:
7656:
7651:
7646:
7644:Control method
7641:
7636:
7631:
7626:
7621:
7615:
7613:
7607:
7606:
7604:
7603:
7598:
7596:Multithreading
7593:
7588:
7583:
7577:
7575:
7569:
7568:
7566:
7565:
7560:
7555:
7550:
7545:
7539:
7537:
7531:
7530:
7528:
7527:
7522:
7517:
7512:
7507:
7502:
7497:
7492:
7490:Formal methods
7487:
7481:
7479:
7473:
7472:
7470:
7469:
7464:
7462:World Wide Web
7459:
7454:
7449:
7444:
7439:
7434:
7429:
7424:
7419:
7414:
7409:
7404:
7398:
7396:
7390:
7389:
7387:
7386:
7381:
7376:
7371:
7366:
7361:
7356:
7351:
7345:
7343:
7336:
7335:
7333:
7332:
7327:
7322:
7317:
7312:
7306:
7304:
7298:
7297:
7295:
7294:
7289:
7284:
7279:
7274:
7269:
7264:
7263:
7262:
7251:
7249:
7243:
7242:
7240:
7239:
7234:
7229:
7224:
7219:
7214:
7209:
7204:
7199:
7194:
7188:
7186:
7180:
7179:
7177:
7176:
7171:
7166:
7161:
7156:
7151:
7146:
7141:
7136:
7131:
7125:
7123:
7113:
7112:
7110:
7109:
7104:
7099:
7094:
7089:
7083:
7081:
7077:
7076:
7074:
7073:
7068:
7063:
7058:
7053:
7048:
7042:
7040:
7034:
7033:
7031:
7030:
7025:
7020:
7015:
7010:
7004:
7002:
6998:
6997:
6995:
6994:
6985:
6980:
6975:
6970:
6965:
6960:
6955:
6950:
6945:
6939:
6937:
6931:
6930:
6923:
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6904:
6897:
6889:
6880:
6879:
6877:
6876:
6864:
6852:
6840:
6825:
6822:
6821:
6819:
6818:
6813:
6808:
6803:
6798:
6793:
6792:
6791:
6784:Mathematicians
6780:
6778:
6776:Related topics
6772:
6771:
6769:
6768:
6763:
6758:
6753:
6748:
6743:
6737:
6735:
6729:
6728:
6726:
6725:
6724:
6723:
6718:
6713:
6711:Control theory
6703:
6698:
6693:
6688:
6683:
6678:
6673:
6668:
6663:
6658:
6653:
6647:
6645:
6639:
6638:
6636:
6635:
6630:
6625:
6620:
6615:
6609:
6607:
6601:
6600:
6598:
6597:
6592:
6587:
6582:
6576:
6574:
6568:
6567:
6565:
6564:
6559:
6554:
6549:
6544:
6539:
6534:
6528:
6526:
6520:
6519:
6517:
6516:
6511:
6506:
6500:
6498:
6492:
6491:
6489:
6488:
6486:Measure theory
6483:
6478:
6473:
6468:
6463:
6458:
6453:
6447:
6445:
6439:
6438:
6436:
6435:
6430:
6425:
6420:
6415:
6410:
6405:
6400:
6394:
6392:
6386:
6385:
6383:
6382:
6377:
6372:
6367:
6362:
6357:
6351:
6349:
6343:
6342:
6340:
6339:
6334:
6329:
6328:
6327:
6322:
6311:
6308:
6307:
6300:
6298:
6297:
6290:
6283:
6275:
6269:
6268:
6263:
6257:
6249:
6246:
6245:
6244:
6231:
6217:
6211:
6205:
6199:
6194:
6184:
6179:
6167:
6153:"Graph theory"
6147:
6146:External links
6144:
6143:
6142:
6136:
6121:
6112:
6099:
6090:
6081:
6077:Academic Press
6068:
6055:
6042:
6024:
6018:
6001:
5992:
5986:
5973:
5964:
5953:
5940:
5937:
5935:
5934:
5927:
5907:
5878:
5865:(3): 491–567,
5849:
5836:(3): 429–490,
5813:
5804:
5768:
5750:
5738:
5715:
5674:
5643:
5637:
5596:
5577:
5560:
5543:
5486:
5416:
5409:
5391:
5374:
5340:
5297:
5278:
5251:(2): 134–143.
5235:
5208:(2): 473–485.
5192:
5171:(1): 1171458.
5148:
5129:(5): 784–786.
5107:
5050:
5013:(1): 113–121.
4997:
4982:
4938:
4936:, p. 161.
4923:
4914:
4912:, p. 149.
4902:
4889:
4887:, p. 148.
4876:
4874:
4871:
4869:
4868:
4863:
4858:
4853:
4848:
4843:
4838:
4836:Sudakov, Benny
4833:
4828:
4823:
4818:
4813:
4808:
4803:
4801:Lovász, László
4798:
4793:
4788:
4783:
4778:
4776:Graham, Ronald
4773:
4768:
4763:
4761:Faudree, Ralph
4758:
4753:
4748:
4743:
4738:
4733:
4728:
4723:
4718:
4716:Bollobás, Béla
4713:
4708:
4701:
4698:
4697:
4696:
4691:
4684:
4681:
4680:
4679:
4674:
4669:
4664:
4657:
4654:
4653:
4652:
4647:
4642:
4637:
4632:
4625:
4622:
4621:
4620:
4615:
4610:
4605:
4600:
4595:
4590:
4585:
4580:
4575:
4570:
4565:
4560:
4555:
4550:
4545:
4540:
4535:
4530:
4523:
4520:
4519:
4518:
4513:
4508:
4503:
4498:
4493:
4488:
4483:
4478:
4473:
4468:
4463:
4458:
4456:Network theory
4453:
4448:
4443:
4438:
4433:
4431:Graph property
4428:
4423:
4418:
4416:Graph equation
4413:
4408:
4403:
4401:Graph database
4398:
4396:Graph coloring
4393:
4388:
4383:
4378:
4373:
4368:
4363:
4361:Data structure
4358:
4353:
4351:Citation graph
4348:
4341:
4340:Related topics
4338:
4337:
4336:
4331:
4326:
4321:
4316:
4310:
4308:
4305:
4304:
4303:
4296:
4285:
4282:
4276:
4265:
4262:
4261:
4260:
4250:
4240:
4234:
4204:
4196:
4193:
4192:
4191:
4188:
4182:
4175:Dominating set
4160:
4157:
4156:
4155:
4148:
4145:
4144:
4143:
4129:
4126:
4125:
4124:
4118:
4113:
4108:
4103:
4098:
4092:
4087:
4080:
4079:Route problems
4077:
4053:
4050:
4049:
4048:
4042:
4036:
4031:, also called
4026:
4021:
4016:
4004:Graph coloring
4002:Main article:
3999:
3998:Graph coloring
3996:
3995:
3994:
3980:
3979:
3975:
3970:complete graph
3946:
3945:
3941:
3936:complete graph
3931:
3894:
3893:
3889:
3878:
3852:
3851:
3850:(NP-complete).
3846:is called the
3824:
3823:
3822:(NP-complete).
3820:clique problem
3791:
3788:
3779:
3776:
3774:
3771:
3759:spanning trees
3728:adjacency list
3703:data structure
3695:Main article:
3692:
3689:
3643:Main article:
3640:
3637:
3631:
3630:Representation
3628:
3551:Wolfgang Haken
3459:
3458:
3364:analysis situs
3358:knight problem
3345:Leonhard Euler
3332:
3329:
3320:
3317:
3300:
3297:
3260:
3257:
3246:or to explore
3220:
3217:
3154:
3151:
3149:, and others.
3131:lattice graphs
3086:
3083:
3028:
3025:
3002:
2999:
2986:
2966:
2942:
2922:
2902:
2899:
2896:
2893:
2890:
2870:
2846:
2822:
2784:
2778:
2774:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2733:
2729:
2726:
2723:
2720:
2700:
2679:
2673:
2669:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2628:
2624:
2621:
2601:
2580:
2576:
2573:
2570:
2556:
2552:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2511:
2490:
2487:
2484:
2481:
2478:
2458:
2434:
2433:
2412:
2408:
2405:
2402:
2388:
2384:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2343:
2339:
2336:
2333:
2330:
2320:
2309:directed lines
2305:directed links
2301:directed edges
2280:
2270:
2242:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2184:directed graph
2176:Multiple edges
2162:
2159:
2156:
2153:
2150:
2137:is called the
2126:
2123:
2120:
2117:
2114:
2094:
2074:
2050:
2030:
2003:
1979:
1955:
1935:
1915:
1895:
1886:directed from
1875:
1872:
1869:
1866:
1863:
1840:
1820:
1817:
1787:
1783:
1767:
1766:
1751:directed lines
1747:directed links
1743:directed edges
1721:
1717:
1714:
1711:
1697:
1693:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1652:
1648:
1645:
1635:
1607:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1555:directed graph
1544:directed graph
1530:Directed graph
1528:Main article:
1525:
1524:Directed graph
1522:
1509:
1506:
1503:
1479:
1459:
1439:
1436:
1433:
1430:
1427:
1407:
1383:
1363:
1340:
1267:
1263:
1259:
1249:of a graph is
1233:
1229:
1225:
1215:of a graph is
1200:
1180:
1156:
1136:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1031:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
949:
929:
926:
923:
920:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
855:
852:
849:
846:
843:
840:
837:
834:
831:
811:
784:
783:
780:unordered pair
763:
760:
757:
754:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
687:
659:
649:
621:
598:
595:
592:
589:
586:
583:
580:
577:
574:
554:
553:
548:
541:
540:
536:
529:
528:
527:
526:
525:
512:multiple edges
498:
478:
454:
434:
407:
387:
367:
364:
361:
358:
355:
337:
336:
304:
301:
298:
295:
283:
280:
277:
274:
271:
268:
265:
262:
259:
256:
253:
250:
247:
244:
234:
206:
183:
180:
177:
174:
171:
168:
165:
138:
135:
120:
117:
37:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8036:
8025:
8022:
8021:
8019:
8004:
8001:
7999:
7996:
7995:
7993:
7991:
7987:
7981:
7978:
7976:
7973:
7971:
7968:
7966:
7963:
7961:
7958:
7956:
7953:
7951:
7948:
7947:
7945:
7943:
7939:
7930:
7925:
7923:
7918:
7916:
7911:
7910:
7907:
7895:
7887:
7885:
7877:
7875:
7867:
7866:
7863:
7857:
7854:
7852:
7849:
7847:
7844:
7842:
7839:
7837:
7834:
7832:
7829:
7827:
7824:
7822:
7819:
7817:
7814:
7812:
7809:
7807:
7804:
7802:
7799:
7797:
7794:
7792:
7789:
7787:
7784:
7782:
7779:
7777:
7774:
7772:
7769:
7767:
7764:
7762:
7759:
7758:
7756:
7752:
7746:
7743:
7741:
7738:
7736:
7733:
7731:
7730:Mixed reality
7728:
7726:
7723:
7721:
7718:
7716:
7713:
7711:
7708:
7707:
7705:
7703:
7699:
7693:
7690:
7688:
7685:
7683:
7680:
7678:
7675:
7673:
7670:
7669:
7667:
7665:
7661:
7655:
7652:
7650:
7647:
7645:
7642:
7640:
7637:
7635:
7632:
7630:
7627:
7625:
7622:
7620:
7617:
7616:
7614:
7612:
7608:
7602:
7599:
7597:
7594:
7592:
7589:
7587:
7584:
7582:
7579:
7578:
7576:
7574:
7570:
7564:
7563:Accessibility
7561:
7559:
7558:Visualization
7556:
7554:
7551:
7549:
7546:
7544:
7541:
7540:
7538:
7536:
7532:
7526:
7523:
7521:
7518:
7516:
7513:
7511:
7508:
7506:
7503:
7501:
7498:
7496:
7493:
7491:
7488:
7486:
7483:
7482:
7480:
7478:
7474:
7468:
7465:
7463:
7460:
7458:
7455:
7453:
7450:
7448:
7445:
7443:
7440:
7438:
7435:
7433:
7430:
7428:
7425:
7423:
7420:
7418:
7415:
7413:
7410:
7408:
7405:
7403:
7400:
7399:
7397:
7395:
7391:
7385:
7382:
7380:
7377:
7375:
7372:
7370:
7367:
7365:
7362:
7360:
7357:
7355:
7352:
7350:
7347:
7346:
7344:
7342:
7337:
7331:
7328:
7326:
7323:
7321:
7318:
7316:
7313:
7311:
7308:
7307:
7305:
7303:
7299:
7293:
7290:
7288:
7285:
7283:
7280:
7278:
7275:
7273:
7270:
7268:
7265:
7261:
7258:
7257:
7256:
7253:
7252:
7250:
7248:
7244:
7238:
7235:
7233:
7230:
7228:
7225:
7223:
7220:
7218:
7215:
7213:
7210:
7208:
7205:
7203:
7200:
7198:
7195:
7193:
7190:
7189:
7187:
7185:
7181:
7175:
7172:
7170:
7167:
7165:
7162:
7160:
7157:
7155:
7152:
7150:
7147:
7145:
7142:
7140:
7137:
7135:
7132:
7130:
7127:
7126:
7124:
7122:
7118:
7114:
7108:
7105:
7103:
7100:
7098:
7095:
7093:
7090:
7088:
7085:
7084:
7082:
7078:
7072:
7069:
7067:
7064:
7062:
7059:
7057:
7054:
7052:
7049:
7047:
7044:
7043:
7041:
7039:
7035:
7029:
7026:
7024:
7021:
7019:
7018:Dependability
7016:
7014:
7011:
7009:
7006:
7005:
7003:
6999:
6993:
6989:
6986:
6984:
6981:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6949:
6946:
6944:
6941:
6940:
6938:
6936:
6932:
6927:
6921:
6917:
6910:
6905:
6903:
6898:
6896:
6891:
6890:
6887:
6875:
6874:
6865:
6863:
6862:
6853:
6851:
6850:
6841:
6839:
6838:
6833:
6827:
6826:
6823:
6817:
6814:
6812:
6809:
6807:
6804:
6802:
6799:
6797:
6794:
6790:
6787:
6786:
6785:
6782:
6781:
6779:
6777:
6773:
6767:
6764:
6762:
6759:
6757:
6754:
6752:
6749:
6747:
6744:
6742:
6739:
6738:
6736:
6734:
6733:Computational
6730:
6722:
6719:
6717:
6714:
6712:
6709:
6708:
6707:
6704:
6702:
6699:
6697:
6694:
6692:
6689:
6687:
6684:
6682:
6679:
6677:
6674:
6672:
6669:
6667:
6664:
6662:
6659:
6657:
6654:
6652:
6649:
6648:
6646:
6644:
6640:
6634:
6631:
6629:
6626:
6624:
6621:
6619:
6616:
6614:
6611:
6610:
6608:
6606:
6602:
6596:
6593:
6591:
6588:
6586:
6583:
6581:
6578:
6577:
6575:
6573:
6572:Number theory
6569:
6563:
6560:
6558:
6555:
6553:
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6529:
6527:
6525:
6521:
6515:
6512:
6510:
6507:
6505:
6504:Combinatorics
6502:
6501:
6499:
6497:
6493:
6487:
6484:
6482:
6479:
6477:
6474:
6472:
6469:
6467:
6464:
6462:
6459:
6457:
6456:Real analysis
6454:
6452:
6449:
6448:
6446:
6444:
6440:
6434:
6431:
6429:
6426:
6424:
6421:
6419:
6416:
6414:
6411:
6409:
6406:
6404:
6401:
6399:
6396:
6395:
6393:
6391:
6387:
6381:
6378:
6376:
6373:
6371:
6368:
6366:
6363:
6361:
6358:
6356:
6353:
6352:
6350:
6348:
6344:
6338:
6335:
6333:
6330:
6326:
6323:
6321:
6318:
6317:
6316:
6313:
6312:
6309:
6304:
6296:
6291:
6289:
6284:
6282:
6277:
6276:
6273:
6267:
6264:
6261:
6258:
6255:
6252:
6251:
6247:
6242:
6238:
6235:
6232:
6229:
6225:
6221:
6218:
6215:
6212:
6209:
6206:
6203:
6200:
6198:
6195:
6192:
6188:
6185:
6183:
6180:
6178:
6174:
6171:
6168:
6164:
6160:
6159:
6154:
6150:
6149:
6145:
6139:
6133:
6129:
6128:
6122:
6118:
6113:
6109:
6108:North-Holland
6105:
6100:
6096:
6091:
6087:
6082:
6078:
6074:
6069:
6065:
6061:
6056:
6049:
6045:
6043:0-13-363473-6
6039:
6032:
6031:
6025:
6021:
6019:0-486-24775-9
6015:
6010:
6009:
6002:
5998:
5993:
5989:
5983:
5979:
5974:
5970:
5965:
5959:
5954:
5950:
5949:
5943:
5942:
5938:
5930:
5924:
5920:
5919:
5911:
5908:
5902:
5897:
5893:
5889:
5882:
5879:
5873:
5868:
5864:
5860:
5853:
5850:
5844:
5839:
5835:
5831:
5824:
5817:
5814:
5808:
5805:
5795:on 2016-03-05
5791:
5784:
5783:
5778:
5772:
5769:
5764:
5760:
5754:
5751:
5741:
5735:
5731:
5730:
5725:
5719:
5716:
5710:
5705:
5701:
5697:
5693:
5689:
5685:
5678:
5675:
5670:
5666:
5662:
5658:
5654:
5647:
5644:
5640:
5638:9780511703690
5634:
5630:
5626:
5622:
5619:, Series IV,
5618:
5617:
5611:
5606:
5600:
5597:
5592:
5588:
5581:
5578:
5573:
5572:
5564:
5561:
5556:
5555:
5547:
5544:
5539:
5535:
5530:
5525:
5521:
5517:
5513:
5509:
5505:
5501:
5497:
5490:
5487:
5482:
5478:
5474:
5470:
5465:
5460:
5455:
5450:
5446:
5442:
5438:
5434:
5427:
5420:
5417:
5412:
5406:
5402:
5395:
5392:
5388:
5384:
5378:
5375:
5364:on 2020-07-28
5360:
5353:
5352:
5344:
5341:
5336:
5332:
5328:
5324:
5320:
5316:
5313:(1): 015102.
5312:
5308:
5301:
5298:
5292:
5291:
5282:
5279:
5274:
5270:
5266:
5262:
5258:
5254:
5250:
5246:
5239:
5236:
5231:
5227:
5223:
5219:
5215:
5211:
5207:
5203:
5196:
5193:
5188:
5184:
5179:
5174:
5170:
5166:
5159:
5152:
5149:
5144:
5140:
5136:
5132:
5128:
5124:
5119:
5111:
5108:
5103:
5099:
5094:
5089:
5085:
5081:
5077:
5073:
5069:
5065:
5061:
5054:
5051:
5046:
5042:
5038:
5034:
5030:
5026:
5021:
5016:
5012:
5008:
5001:
4998:
4993:
4989:
4985:
4983:9781450326223
4979:
4975:
4971:
4967:
4963:
4958:
4953:
4949:
4942:
4939:
4935:
4930:
4928:
4924:
4918:
4915:
4911:
4906:
4903:
4899:
4893:
4890:
4886:
4881:
4878:
4872:
4867:
4864:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4846:Thomas, Robin
4844:
4842:
4839:
4837:
4834:
4832:
4831:Seymour, Paul
4829:
4827:
4824:
4822:
4819:
4817:
4816:Rényi, Alfréd
4814:
4812:
4809:
4807:
4804:
4802:
4799:
4797:
4794:
4792:
4791:Kotzig, Anton
4789:
4787:
4784:
4782:
4781:Harary, Frank
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4711:Berge, Claude
4709:
4707:
4704:
4703:
4699:
4695:
4692:
4690:
4687:
4686:
4682:
4678:
4677:Ramsey theory
4675:
4673:
4670:
4668:
4665:
4663:
4662:Combinatorics
4660:
4659:
4655:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4627:
4623:
4619:
4616:
4614:
4611:
4609:
4606:
4604:
4601:
4599:
4596:
4594:
4591:
4589:
4586:
4584:
4581:
4579:
4576:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4554:
4551:
4549:
4546:
4544:
4541:
4539:
4536:
4534:
4531:
4529:
4526:
4525:
4521:
4517:
4514:
4512:
4509:
4507:
4504:
4502:
4499:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4481:Quantum graph
4479:
4477:
4476:Perfect graph
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4447:
4446:Logical graph
4444:
4442:
4441:Knight's Tour
4439:
4437:
4434:
4432:
4429:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4411:Graph drawing
4409:
4407:
4404:
4402:
4399:
4397:
4394:
4392:
4389:
4387:
4386:Graph algebra
4384:
4382:
4379:
4377:
4374:
4372:
4369:
4367:
4364:
4362:
4359:
4357:
4354:
4352:
4349:
4347:
4344:
4343:
4339:
4335:
4332:
4330:
4327:
4325:
4322:
4320:
4317:
4315:
4312:
4311:
4306:
4301:
4297:
4294:
4290:
4286:
4283:
4281:
4277:
4274:
4271:
4270:
4269:
4264:Graph classes
4263:
4258:
4257:regular graph
4254:
4251:
4248:
4244:
4243:Edge coloring
4241:
4238:
4235:
4232:
4229:
4228:
4227:
4224:
4220:
4213:
4207:
4203:
4194:
4189:
4186:
4183:
4180:
4179:neighborhoods
4176:
4173:
4172:
4171:
4169:
4165:
4158:
4154:
4151:
4150:
4146:
4142:
4139:
4138:
4137:
4135:
4127:
4122:
4119:
4117:
4114:
4112:
4109:
4107:
4104:
4102:
4099:
4096:
4093:
4091:
4088:
4086:
4083:
4082:
4078:
4076:
4074:
4070:
4066:
4065:compositional
4061:
4059:
4058:partial order
4051:
4046:
4043:
4040:
4037:
4034:
4030:
4027:
4025:
4022:
4020:
4017:
4015:
4012:
4011:
4010:
4005:
3997:
3993:
3989:
3988:
3987:
3985:
3974:
3971:
3967:
3963:
3959:
3955:
3954:
3953:
3951:
3940:
3937:
3930:
3927:
3923:
3919:
3918:
3917:
3915:
3911:
3907:
3906:homeomorphism
3903:
3899:
3888:
3884:
3877:
3874:
3870:
3866:
3865:
3864:
3862:
3857:
3849:
3845:
3841:
3840:
3839:
3836:
3831:
3829:
3821:
3817:
3816:
3815:
3813:
3809:
3805:
3801:
3797:
3789:
3787:
3785:
3777:
3772:
3770:
3768:
3767:shortest path
3764:
3760:
3756:
3752:
3748:
3744:
3743:degree matrix
3740:
3736:
3731:
3729:
3725:
3720:
3717:
3713:
3712:sparse graphs
3708:
3704:
3698:
3690:
3688:
3686:
3682:
3678:
3673:
3671:
3667:
3662:
3660:
3655:
3651:
3646:
3645:Graph drawing
3638:
3636:
3629:
3627:
3625:
3624:
3619:
3615:
3610:
3608:
3604:
3600:
3596:
3592:
3588:
3584:
3580:
3576:
3571:
3569:
3565:
3561:
3557:
3552:
3548:
3547:Kenneth Appel
3544:
3539:
3537:
3536:
3531:
3527:
3523:
3519:
3515:
3511:
3507:
3503:
3499:
3495:
3491:
3488:addressed to
3487:
3483:
3479:
3474:
3472:
3468:
3464:
3456:
3452:
3448:
3444:
3440:
3439:
3438:
3436:
3435:
3430:
3425:
3422:
3418:
3414:
3410:
3406:
3405:
3400:
3396:
3392:
3388:
3383:
3381:
3377:
3373:
3369:
3366:initiated by
3365:
3361:
3359:
3354:
3350:
3346:
3337:
3330:
3328:
3326:
3318:
3316:
3314:
3310:
3306:
3298:
3296:
3294:
3289:
3286:
3282:
3278:
3274:
3269:
3266:
3258:
3256:
3253:
3249:
3245:
3241:
3233:
3230:
3225:
3218:
3216:
3214:
3209:
3205:
3201:
3196:
3192:
3188:
3184:
3180:
3176:
3172:
3168:
3164:
3160:
3152:
3150:
3148:
3144:
3140:
3136:
3132:
3129:, which uses
3128:
3124:
3120:
3116:
3112:
3108:
3104:
3100:
3096:
3092:
3084:
3082:
3080:
3076:
3072:
3068:
3064:
3060:
3056:
3052:
3047:
3043:
3038:
3034:
3026:
3024:
3022:
3017:
3007:
3000:
2998:
2984:
2964:
2956:
2940:
2920:
2897:
2894:
2891:
2868:
2860:
2844:
2836:
2820:
2811:
2809:
2808:
2803:
2799:
2782:
2776:
2772:
2768:
2762:
2759:
2756:
2750:
2744:
2741:
2738:
2731:
2724:
2721:
2718:
2698:
2677:
2671:
2667:
2663:
2657:
2654:
2651:
2645:
2639:
2636:
2633:
2626:
2622:
2619:
2599:
2578:
2574:
2571:
2568:
2554:
2550:
2546:
2540:
2537:
2534:
2528:
2522:
2519:
2516:
2509:
2485:
2482:
2479:
2456:
2448:
2447:
2441:
2439:
2431:
2427:
2410:
2406:
2403:
2400:
2386:
2382:
2378:
2372:
2369:
2366:
2360:
2354:
2351:
2348:
2341:
2334:
2331:
2328:
2321:
2318:
2314:
2310:
2306:
2302:
2299:(also called
2298:
2294:
2278:
2271:
2268:
2264:
2261:(also called
2260:
2256:
2240:
2233:
2232:
2231:
2214:
2211:
2208:
2205:
2202:
2196:
2193:
2185:
2180:
2178:
2177:
2157:
2154:
2151:
2140:
2139:inverted edge
2121:
2118:
2115:
2092:
2072:
2064:
2048:
2028:
2021:
2017:
2001:
1993:
1977:
1970:of the edge,
1969:
1953:
1933:
1913:
1893:
1870:
1867:
1864:
1852:
1838:
1818:
1815:
1807:
1785:
1781:
1772:
1764:
1763:ordered pairs
1760:
1756:
1752:
1748:
1744:
1741:(also called
1740:
1736:
1719:
1715:
1712:
1709:
1695:
1691:
1687:
1681:
1678:
1675:
1669:
1663:
1660:
1657:
1650:
1646:
1643:
1636:
1633:
1629:
1626:(also called
1625:
1621:
1605:
1598:
1597:
1596:
1579:
1576:
1573:
1567:
1564:
1556:
1551:
1549:
1545:
1536:
1531:
1523:
1521:
1507:
1504:
1501:
1493:
1477:
1457:
1434:
1431:
1428:
1405:
1397:
1381:
1361:
1354:
1338:
1329:
1316:
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6547:Differential
6514:Order theory
6509:Graph theory
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6220:Online books
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3200:porous media
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3088:
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59:graph theory
58:
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8003:Mathematica
7990:Proprietary
7836:Video games
7816:Digital art
7573:Concurrency
7442:Data mining
7354:Probability
7087:Interpreter
6873:WikiProject
6716:Game theory
6696:Probability
6433:Homological
6423:Multilinear
6403:Commutative
6380:Type theory
6347:Foundations
6303:mathematics
5724:Tutte, W.T.
4751:Erdős, Paul
4672:Knot theory
4273:Enumerating
4249:as possible
3966:subdivision
3964:contains a
3920:A graph is
3902:subdivision
3898:subdivision
3867:A graph is
3778:Enumeration
3659:W. T. Tutte
3471:Pólya Prize
3463:Dénes Kőnig
3353:Vandermonde
3305:knot theory
3299:Mathematics
3091:linguistics
3085:Linguistics
3071:transaction
1122:pseudograph
1120:undirected
788:undirected
119:Definitions
55:mathematics
7894:Glossaries
7766:E-commerce
7359:Statistics
7302:Algorithms
7260:Stochastic
7092:Middleware
6948:Peripheral
6701:Statistics
6580:Arithmetic
6542:Arithmetic
6408:Elementary
6375:Set theory
5939:References
5799:2016-03-14
5745:2016-03-14
5605:Cayley, A.
5593:: 169–189.
5368:2019-10-30
4856:Turán, Pál
4736:Chung, Fan
4706:Alon, Noga
4689:Hypergraph
4522:Algorithms
4461:Null graph
4289:algorithms
4231:Arboricity
4047:(unsolved)
4041:(unsolved)
3808:hereditary
3579:Kuratowski
3389:and while
3387:Königsberg
3181:and edges
3173:summarize
3139:TextGraphs
3113:. Within
3075:persistent
3051:algorithms
2061:and to be
790:multigraph
465:and to be
7950:Cytoscape
7715:Rendering
7710:Animation
7341:computing
7292:Semantics
6983:Processor
6628:Geometric
6618:Algebraic
6557:Euclidean
6532:Algebraic
6428:Universal
6163:EMS Press
6012:. Dover.
5520:0006-8950
5481:214722561
5473:2475-9066
5335:0021-8979
5245:Neurology
5187:114999767
5143:0018-9219
5084:0006-8950
4957:1312.0976
4247:matchings
4123:(NP-hard)
3910:planarity
3881:(see the
3863:states:
3724:edge list
3707:algorithm
3556:Robertson
3486:De Morgan
3429:Sylvester
3421:De Bruijn
3409:chemistry
3376:L'Huilier
3240:sociology
3232:Sociogram
3206:uses the
3159:chemistry
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2699:ϕ
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297:≠
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246:⊆
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7975:NetworkX
7955:Graphviz
7874:Category
7702:Graphics
7477:Security
7139:Compiler
7038:Networks
6935:Hardware
6849:Category
6605:Topology
6552:Discrete
6537:Analytic
6524:Geometry
6496:Discrete
6451:Calculus
6443:Analysis
6398:Abstract
6337:Glossary
6320:Timeline
6237:Archived
6173:Archived
6048:Archived
5894:: 2–44,
5761:(1992),
5726:(2001),
5607:(1857),
5538:31099821
5273:28334693
5265:23719145
5222:26960946
5102:31099821
4992:14027025
4624:Subareas
4307:See also
4298:Finding
3934:nor the
3916:states:
3800:subgraph
3773:Problems
3587:topology
3522:Petersen
3510:Hadwiger
3490:Hamilton
3447:Kekuléan
3380:topology
3277:genomics
2955:adjacent
2259:vertices
2063:incident
1624:vertices
1492:adjacent
638:vertices
467:incident
223:vertices
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7884:Outline
6861:Commons
6643:Applied
6613:General
6390:Algebra
6315:History
6189:at the
6165:, 2001
5696:Bibcode
5529:6598625
5449:bioRxiv
5441:Bibcode
5315:Bibcode
5230:3987492
5093:6598625
5045:9233932
5025:Bibcode
4962:Bibcode
4223:edges.
3751:degrees
3603:current
3599:voltage
3583:Whitney
3564:Sanders
3560:Seymour
3502:Heawood
3391:Listing
3368:Leibniz
3355:on the
3347:on the
3331:History
3285:pathway
3265:biology
3259:Biology
3234:(1953).
3163:physics
3147:VerbNet
3143:WordNet
3073:-safe,
3042:website
3031:Within
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2085:and on
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3506:Ramsey
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3229:Moreno
3095:syntax
3037:causal
2807:quiver
2804:(or a
2800:and a
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1806:tuples
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1289:degree
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646:points
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7287:Logic
7121:tools
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6305:areas
6051:(PDF)
6034:(PDF)
5963:2001.
5826:(PDF)
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1258:|
1232:|
1228:V
1224:|
1199:E
1179:V
1155:E
1135:V
1098:}
1095:V
1089:y
1086:,
1083:x
1077:}
1074:y
1071:,
1068:x
1065:{
1062:{
1056:E
1053::
1010:}
1007:V
1001:y
998:,
995:x
989:}
986:y
983:,
980:x
977:{
974:{
968:E
948:E
928:}
925:y
919:x
907:V
901:y
898:,
895:x
889:}
886:y
883:,
880:x
877:{
874:{
854:}
851:x
848:{
845:=
842:}
839:x
836:,
833:x
830:{
810:x
762:}
759:y
753:x
741:V
735:y
732:,
729:x
723:}
720:y
717:,
714:x
711:{
708:{
702:E
699::
658:E
620:V
597:)
591:,
588:E
585:,
582:V
579:(
576:=
573:G
497:y
477:x
453:y
433:x
406:y
386:x
366:}
363:y
360:,
357:x
354:{
303:}
300:y
294:x
282:V
276:y
273:,
270:x
264:}
261:y
258:,
255:x
252:{
249:{
243:E
205:V
182:)
179:E
176:,
173:V
170:(
167:=
164:G
35:.
20:)
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