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| graph (mathematics) | |
|---|---|
| Name | Graph (mathematics) |
| Field | Mathematics |
| Introduced | 1736 |
| Key people | Leonhard Euler, Königsberg bridge problem, Arthur Cayley, Gustav Kirchhoff |
graph (mathematics) is a combinatorial structure consisting of vertices connected by edges used to model pairwise relations between objects. It appears across Leonhard Euler's work on the Königsberg bridges, in Arthur Cayley's enumeration of trees, and in applications spanning Gustav Kirchhoff's circuit theory, Paul Erdős's extremal problems, and modern computational studies at institutions like Massachusetts Institute of Technology and Stanford University. Graphs provide a unifying language for problems studied by researchers associated with Royal Society, Princeton University, University of Cambridge, and industrial labs such as Bell Labs.
A graph comprises a set of vertices (or nodes) and a set of edges that join pairs of vertices; formalizations appear in texts by Dénes Kőnig, Frank Harary, and Claude Shannon. Terms include degree, path, cycle, adjacency, incidence, and connectivity, discussed in works from Cambridge University Press and by scholars at Oxford University. Related notions—multigraph, pseudograph, directed edge, loop, and labeled versus unlabeled graphs—are treated in classic monographs by Ronald Graham and Miklós Bóna.
Common families include simple graphs, directed graphs (digraphs) studied by Kazimierz Kuratowski and Philip Hall, weighted graphs used in Dijkstra's algorithm, bipartite graphs central to results by Konrad Zuse and Kőnig's theorem, planar graphs appearing in Kuratowski's characterization and in work related to the Four Color Theorem, and hypergraphs introduced in combinatorial set theory by researchers at University of Chicago. Other classes include multigraphs, infinite graphs examined by Paul Erdős, and random graphs modeled by Erdős–Rényi processes developed with Alfréd Rényi.
Key invariants include degree sequences, chromatic number (central to the Four Color Theorem and work by Kenneth Appel and Wolfgang Haken), clique number, independence number, chromatic index (studied in Vizing's theorem), girth, diameter, radius, and eigenvalues of adjacency or Laplacian matrices used in spectral graph theory by Fan Chung and László Lovász. Connectivity, edge-connectivity, planarity characterized by Kuratowski and Wagner, and matching number linked to Edmonds's blossom algorithm are fundamental. Graph minors form the basis of the Robertson–Seymour theory proved by Neil Robertson and Paul D. Seymour.
Classic examples include complete graphs (K_n), cycles (C_n), paths (P_n), trees as in Arthur Cayley's enumeration, bipartite complete graphs K_{m,n} related to Kőnig's theorem, and Petersen graph studied by Fritz Petersen. Other notable examples are planar graphs like platonic solids studied by Euclid-inspired geometry literature, Cayley graphs linking group theory work by Arthur Cayley to geometric group theory, de Bruijn graphs used in coding theory and genomics associated with labs at Broad Institute, and expander graphs constructed in work by Alon and Margulis.
Standard operations include union, join, complement, Cartesian product, tensor product (Kronecker product) linked to linear algebra texts from Institute of Electrical and Electronics Engineers, and line graph construction studied by Harary. Contraction and deletion underpin the Tutte polynomial developed by W. T. Tutte and relate to matroid theory explored at University of Waterloo. Graph drawing routines and embeddings into surfaces are connected to topological studies at Princeton University and algorithms by researchers at Bell Labs.
Fundamental algorithms include search methods such as depth-first search and breadth-first search from early computer science curricula at Massachusetts Institute of Technology, shortest path algorithms by Edsger Dijkstra and Dijkstra's contemporaries, minimum spanning tree algorithms by Kruskal and Prim, maximum flow by Ford–Fulkerson and refinements by Edmonds–Karp, and matching algorithms by Jack Edmonds. Complexity results relate to Cook's theorem, Stephen Cook's work on NP-completeness, and reductions used in Richard Karp's list of NP-complete problems. Approximation algorithms and fixed-parameter tractability have been advanced by researchers affiliated with European Research Council grants and centers at ETH Zurich.
Graphs model networks in studies at AT&T and Google, inform social network analysis rooted in work by Geoffrey West-adjacent scholars, underpin chemistry via chemical graph theory influenced by August Kekulé-related history, and support bioinformatics applications developed at National Institutes of Health and Broad Institute. Historically, the field traces from Leonhard Euler's 1736 solution to the Königsberg bridge problem through 19th-century work by Arthur Cayley, 20th-century foundations by Dénes Kőnig and Frank Harary, to modern advances by Paul Erdős, László Lovász, and the Robertson–Seymour project; ongoing research continues at universities such as Stanford University and institutes like Institute for Advanced Study.