graph (discrete mathematics)

graph (discrete mathematics)
Name	Graph (discrete mathematics)
Caption	Example of a simple undirected graph
Field	Discrete mathematics, Combinatorics, Computer science
Introduced	1736
Notable	Leonhard Euler, Paul Erdős, Klaus Wagner

graph (discrete mathematics) is a mathematical structure used to model pairwise relations between objects. A graph comprises a set of vertices and a set of edges connecting pairs of vertices; it appears across work by Leonhard Euler, Arthur Cayley, Paul Erdős, Kurt Gödel, Alan Turing and in institutions such as the Bell Labs, Massachusetts Institute of Technology, Princeton University, and University of Cambridge. Graph theory bridges problems studied in the Seven Bridges of Königsberg, the Four Color Theorem, the P versus NP problem, Gödel, Escher, Bach, and modern results associated with the Erdős–Rényi model.

Definitions and basic concepts

A graph G is defined by a vertex set V(G) and an edge set E(G), where edges are unordered or ordered pairs; these foundations were influenced by Leonhard Euler and formalized in texts by Dénes Kőnig and Frank Harary. Terms include degree, adjacency, path, cycle, connectedness and components, each appearing in the work of Kazimierz Kuratowski, Klaus Wagner, Wacław Sierpiński, and researchers at University of Oxford and Harvard University. Variants introduce loops, multiple edges, and labeled or unlabeled vertices — ideas appearing in enumerative results by Arthur Cayley, combinatorial enumerations by Paul Erdős and analytic methods used by Andrey Kolmogorov.

Types of graphs

Common classes include simple graphs, multigraphs, directed graphs (digraphs), weighted graphs, bipartite graphs, complete graphs, regular graphs, planar graphs and hypergraphs; studies of planarity invoked work by Kuratowski's theorem linked to Klaus Wagner and contributed to proofs related to the Four Color Theorem explored by researchers at University of Illinois Urbana–Champaign and University of Cambridge. Special families such as trees, forests, cacti, chordal graphs, interval graphs, comparability graphs, permutation graphs and Cayley graphs feature in research by Arthur Cayley, Richard Stanley, László Lovász, and networks studied at Bell Labs and IBM Research. Random graph models include the Erdős–Rényi model, preferential attachment models related to work by Duncan J. Watts and Albert-László Barabási, and spatial networks tied to studies at Los Alamos National Laboratory.

Graph representations and matrices

Graphs are encoded by adjacency matrices, incidence matrices, Laplacian matrices and distance matrices; spectral graph theory relates eigenvalues to structure in research by Fan Chung, Mihran Katz, and institutions like Princeton University and Stanford University. The graph Laplacian underpins connections to the Heat equation and spectral clustering techniques championed by groups at Google and Microsoft Research. Data structures such as adjacency lists, edge lists and compressed sparse row formats are central in implementations used at Bell Labs and in software from IBM Research and projects associated with LINUX Foundation and Apache Software Foundation.

Fundamental properties and invariants

Invariants include chromatic number, clique number, independence number, girth, diameter, radius, connectivity, matching number and eigenvalue spectra; major theorems and conjectures involve contributions by Paul Erdős, László Lovász, Klaus Wagner, Dénes Kőnig, Egon Balas and developments at Institute for Advanced Study. Theorems such as Turán's theorem, Ramsey theory, Menger's theorem, Hall's marriage theorem and König's theorem link to combinatorial optimization research at Princeton University, University of Cambridge and École Normale Supérieure. Extremal graph theory and probabilistic methods originate from work by Paul Erdős and collaborators at venues like Jerusalem's Hebrew University.

Graph algorithms and complexity

Algorithms for traversal, shortest paths, minimum spanning trees, matchings, network flows, coloring, and isomorphism are foundational; classical algorithms include those of Edsger W. Dijkstra, Kruskal's algorithm associated with Joseph Kruskal, Edmonds' blossom algorithm from Jack Edmonds, and the Ford–Fulkerson method studied at University of California, Berkeley. Complexity classifications reference the P versus NP problem, reductions by Stephen Cook, hardness results influenced by Richard Karp and ongoing research in isomorphism at National Institute of Standards and Technology and Microsoft Research. Modern algorithmic graph theory is applied in parallel and distributed settings by teams at Google, Facebook, Amazon and research groups at MIT and Carnegie Mellon University.

Applications and examples

Graphs model networks in sociology, biology, telecommunications, transportation and chemistry, shaping studies at Harvard Medical School, Centers for Disease Control and Prevention, NASA, European Space Agency, Siemens and Bayer. Examples include social networks analyzed by teams at Facebook and Stanford University, protein–protein interaction networks researched at Cold Spring Harbor Laboratory and Broad Institute, circuit designs by engineers at Intel and ARM Holdings, and route planning systems developed by Google Maps and Uber. Theoretical applications connect to coding theory at Bell Labs and AT&T, cryptographic constructions informed by researchers at National Security Agency, and economic networks studied at London School of Economics.

Category:Graph theory