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Graph with almost the max amount of edges
In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected
Dense_graph
Function type in graph theory
important in the study of dense graphs. Graphons arise both as a natural notion for the limit of a sequence of dense graphs, and as the fundamental defining
Graphon
Graph where every edge is in one triangle
problem. Although dense graphs can have a number of edges proportional to the square of the number of vertices, locally linear graphs have a smaller number
Locally_linear_graph
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes
Glossary_of_graph_theory
Least-weight tree connecting graph vertices
graph theory, a minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that
Minimum_spanning_tree
Area of discrete mathematics
computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context
Graph_theory
Directed graph with no directed cycles
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it
Directed_acyclic_graph
Node labeling problem in graph theory
In graph theory, the graph bandwidth problem may be visualized as placing the vertices of a given graph at distinct integer positions along the number
Graph_bandwidth
Adjacent subset of an undirected graph
size of a clique in dense graphs. If a graph has sufficiently many edges, it must contain a large clique. For instance, every graph with n {\displaystyle
Clique_(graph_theory)
Topic in computer science
Many interesting properties of dense graphs can be tested using query complexity that depends only on ε and not on the graph size n. However, the query complexity
Property_testing
vertices in a dense graph such that every small subset of vertices has many common neighbors. It is a useful tool to embed a graph into another graph with many
Dependent_random_choice
Graph that can be embedded in the plane
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect
Planar_graph
Graph partition into regular subgraphs
certain properties of random graphs can be applied to dense graphs like counting the copies of a given subgraph within graphs. Endre Szemerédi proved the
Szemerédi_regularity_lemma
Bruijn graph Dense graph Dipole graph Directed acyclic graph Directed graph Distance regular graph Distance-transitive graph Edge-transitive graph Interval
List_of_graph_theory_topics
Influence of local substructure of a graph on global properties
various graphs H {\displaystyle H} . By extending the homomorphism density to graphons, which are objects that arise as a limit of dense graphs, the graph homomorphism
Extremal_graph_theory
Abstract data type in computer science
science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within
Graph_(abstract_data_type)
Subgraph induced by all nodes linked to a given node of a graph
In graph theory, the neighbourhood of a vertex v in a graph G is the subgraph of G induced by all the vertices that are connected to v by an edge (vertices
Neighbourhood_(graph_theory)
Highly connected subgraph
In graph theory and computer science, a dense subgraph is a subgraph with many edges per vertex. This is formalized as follows: let G = (V, E) be an undirected
Dense_subgraph
Graph without triples of adjacent vertices
area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently
Triangle-free_graph
Hungarian mathematician (born 1943)
the structure of dense graphs; he was the first to prove detailed results about the phase transition in the evolution of random graphs; he proved that
Béla_Bollobás
Algorithm for finding shortest paths
an algorithm for finding the shortest paths between nodes in a weighted graph, which may represent, for example, a road network. It was conceived by computer
Dijkstra's_algorithm
Computational problem of graph theory
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights
Shortest_path_problem
Similarly, in the context of graph theory, if the number of links is close to its maximum, then the graph would be known as dense graph. If the number of links
Sparse_network
Computer science algorithm
been explored. As graphs become more dense, this redundancy becomes more prevalent, causing computation time to increase; as graphs become more sparse
Graph_traversal
Regular graph with fewest possible nodes for its girth
of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an (r, g)-graph is defined to be a graph in which
Cage_(graph_theory)
Award for advancements in discrete mathematics
Balázs Szegedy for characterizing subgraph multiplicity in sequences of dense graphs. 2015: Francisco Santos Leal for a counter-example of the Hirsch conjecture
Fulkerson_Prize
Large connected component of a random graph
graphs, a giant component is a connected component whose fraction of the overall number of vertices is bounded away from zero. In sufficiently dense graphs
Giant_component
Geometric graph with unit edge lengths
In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting
Unit_distance_graph
Form of data structure
A scene graph is a hierarchical data structure commonly used by vector-based graphics editing applications and modern computer games, which cascades the
Scene_graph
Edges that hit all cycles in a graph
In graph theory and graph algorithms, a feedback arc set or feedback edge set in a directed graph is a subset of the edges of the graph that contains at
Feedback_arc_set
Generalization of graph theory
hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two
Hypergraph
Concerned with the notion of stability in model theory
of any nowhere dense graph class. These include graph classes with bounded expansion, which in turn include planar graphs and any graph class of bounded
Stable_theory
Algorithm for the directed version of the minimum spanning tree problem
V ) {\displaystyle O(E\log V)} for sparse graphs and O ( V 2 ) {\displaystyle O(V^{2})} for dense graphs. This is as fast as Prim's algorithm for an
Edmonds'_algorithm
combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Graph coloring related to treedepth
In graph theory, a centered coloring is a type of graph coloring related to treedepth. The minimum number of colors in a centered coloring of a graph equals
Centered_coloring
On graph drawing with integer edge lengths
graphs with rational edge lengths (and therefore, after scaling them appropriately, with integer edge lengths). However, Ulam conjectured that dense rational-distance
Harborth's_conjecture
Graph with same nodes as but complementary connections to another
In the mathematical field of graph theory, the complement or inverse of a graph G is a graph H on the same vertices such that two distinct vertices are
Complement_graph
belongs to a unique triangle; that is, it is locally linear. Finding large dense graphs with this property is one of the formulations of the Ruzsa–Szemerédi
Brouwer–Haemers_graph
Randomized algorithm for minimum cuts
In computer science and graph theory, Karger's algorithm is a randomized algorithm to compute a minimum cut of a connected graph. It was invented by David
Karger's_algorithm
Algorithm for finding the shortest paths in graphs
iteration shrinks, leading to a constant-factor savings in time for dense graphs. This variation can be implemented by keeping a collection of vertices
Bellman–Ford_algorithm
Algorithm for maximum cardinality matching
(|V|)} . In the case of dense graphs the time bound becomes O ( | V | 2.5 ) {\displaystyle O(|V|^{2.5})} , and for sparse random graphs it runs in time O (
Hopcroft–Karp_algorithm
Property in graph theory
n-vertex t-biclique-free graph has O(n2 − 1/t) edges, significantly fewer than a dense graph would have. Conversely, if a graph family is defined by forbidden
Biclique-free_graph
Algorithm in graph theory
Floyd-Warshall algorithm, the constants involved matter quite a lot. When a graph is dense (i.e., | E | ≈ | V | 2 {\displaystyle |E|\approx |V|^{2}} ), the Floyd-Warshall
Floyd–Warshall_algorithm
Subset of a graph's vertices, including at least one endpoint of every edge
In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In
Vertex_cover
Algebraic encoding of graph connectivity
is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph G {\displaystyle
Tutte_polynomial
Triangle-free graph requiring four colors
In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number
Grötzsch_graph
Logical formulation of graph properties
the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using sentences
Logic_of_graphs
Measure of graph complexity
closely related to treewidth, but unlike treewidth it can be small for dense graphs. It is defined as the minimum number of labels needed to construct G
Clique-width
Method for finding minimum spanning trees
asymptotically faster when the graph is dense enough that |E| is ω(|V|), and linear time when |E| is at least |V| log |V|. For graphs of even greater density
Prim's_algorithm
The twin-width of a graph is the smallest value of d {\displaystyle d} for which it has a d {\displaystyle d} -sequence. A dense graph may still have bounded
Twin-width
Type of ordering of a set
partial order or total order < on a set X {\displaystyle X} is said to be dense if, for all x {\displaystyle x} and y {\displaystyle y} in X {\displaystyle
Dense_order
Problem on triangles in graph theory
also known to hold for graphs of treewidth at most six, for threshold graphs, for sufficiently dense graphs, and for chordal graphs that don't contain a
Tuza's_conjecture
Problem in theoretical computer science
the subgraph isomorphism problem is a computational task in which two graphs G {\displaystyle G} and H {\displaystyle H} are given as input, and one
Subgraph_isomorphism_problem
Property in graph theory
In graph theory, the cutwidth of an undirected graph is the smallest integer k {\displaystyle k} with the following property: there is an ordering of
Cutwidth
American mathematician
characterized uniformly dense graphs, and have found several classes of uniformly dense graphs and several ways of constructing such graphs. Hobbs had also done
Arthur_Hobbs_(mathematician)
Fiber-optic communications technology
transmission windows of silica fibers. Dense WDM (DWDM) uses the C-Band (1530 nm-1565 nm) transmission window but with denser channel spacing. Channel plans vary
Wavelength-division multiplexing
Wavelength-division_multiplexing
Graph theory problem: find a matching containing the most edges
algorithm for general graphs with complexity O ( V 2.372 ) {\displaystyle O(V^{2.372})} . This is better in theory for sufficiently dense graphs, but in practice
Maximum-cardinality_matching
Concept in extremal graph theory
In graph theory, an area of mathematics, common graphs belong to a branch of extremal graph theory concerning inequalities in homomorphism densities. Roughly
Common_graph
graphs under the correct conditions. The lemma allowed for deeper exploration into the nature of embeddings of large sparse graphs into dense graphs.
Gábor_N._Sárközy
Norwegian mathematician (1899–1968)
on their applications. Within graph theory, Ore's theorem is one of several results proving that sufficiently dense graphs contain Hamiltonian cycles. Ore
Øystein_Ore
Hungarian mathematician (born 1948)
July 17, 1948) is a Hungarian mathematician known for her contributions to graph theory and discrete geometry. A student of Vera T. Sós and a co-author of
Katalin_Vesztergombi
Structure-preserving correspondence between node-link graphs
In the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a
Graph_homomorphism
mathematics and extremal graph theory, the Ruzsa–Szemerédi problem or (6,3)-problem asks for the maximum number of edges in a graph in which every edge belongs
Ruzsa–Szemerédi_problem
Graph linking pairs of comparable elements in a partial order
Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs, and divisor graphs. An incomparability
Comparability_graph
Theorem in functional analysis
norm is essential in designing efficient approximation algorithms for dense graphs and matrices. More generally, the definition of cut norm can be generalized
Grothendieck_inequality
French mathematician
properties on nowhere dense structures, and On nowhere dense graphs. With Jaroslav Nešetřil, he is the author of the book Sparsity: Graphs, Structures, and
Patrice_Ossona_de_Mendez
Graph without four-vertex star subgraphs
In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the
Claw-free_graph
Independent set in a graph
In graph theory, a rainbow-independent set (ISR) is an independent set in a graph, in which each vertex has a different color. Formally, let G = (V, E)
Rainbow-independent_set
Data structure that maintains info about the connected components of a graph
graph theory, a dynamic connectivity structure is a data structure that dynamically maintains information about the connected components of a graph.
Dynamic_connectivity
Graph minor formed from subgraphs of small diameter
dense class of graphs, then (for every ε > 0) the n-vertex graphs in F have O(n1 + ε) edges; thus, the nowhere dense graphs are sparse graphs. A more restrictive
Shallow_minor
Networks with multiple kinds of relations
Zhou (2006). "CLAN: An Algorithm for Mining Closed Cliques from Large Dense Graph Databases" (PDF). 22nd International Conference on Data Engineering (ICDE'06)
Multidimensional_network
Ensemble of states at a constant temperature
Roccaverde, A.; Starreveld, N. J. (2018). "Ensemble equivalence for dense graphs". Electronic Journal of Probability. 23. arXiv:1703.08058. doi:10.1214/18-EJP135
Canonical_ensemble
Topics referred to by the same term
density in differential geometry Density (graph theory), the fraction of possible edges that exist in a graph Density (computer storage), a measure of
Density_(disambiguation)
Measurement of graph sparsity
In graph theory, a k-degenerate graph is an undirected graph in which every non-empty subgraph has at least one vertex of degree at most k {\displaystyle
Degeneracy_(graph_theory)
Logic problem, AND of pairwise ORs
First published by Cheriyan, J.; Mehlhorn, K. (1996), "Algorithms for dense graphs and networks on the random access computer", Algorithmica, 15 (6): 521–549
2-satisfiability
Copy of a directed graph with redundant edges removed
In the mathematical field of graph theory, a transitive reduction of a directed graph D is another directed graph with the same vertices and as few edges
Transitive_reduction
However, graphs whose incidence posets have order dimension 4 may be dense and may have unbounded chromatic number. Every complete graph on n vertices
Incidence_poset
Israeli mathematician
רפאל יוסטר) is an Israeli mathematician specializing in combinatorics and graph theory. He is a professor of mathematics at the University of Haifa. He
Raphael_Yuster
Independent set which is not a subset of any other independent set
In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other
Maximal_independent_set
Type of graph in mathematics and physics
mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph) in which each edge is given a
Quantum_graph
Computational problem in graph theory
November 2020). Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs. Durham, NC, USA: IEEE. pp. 919–930. Brand, J. vd; Lee, Y.T.; Liu, Y
Maximum_flow_problem
Number of forests a graph's edges may be partitioned into
of a graph is a measure of how dense the graph is: graphs with many edges have high arboricity, and graphs with high arboricity must have a dense subgraph
Arboricity
Region of Earth surrounding the Equator
Graph showing the zonally averaged monthly precipitation. The tropics receive more precipitation than higher latitudes. The precipitation maximum, which
Tropics
On chains and antichains in partial orders
comparability graph is itself a comparability graph, formed from the restriction of the partial order to a subset of its elements. An undirected graph is perfect
Dilworth's_theorem
Czech mathematician (born 1946)
dense structures, and On nowhere dense graphs. Scholia has a profile for Jaroslav Nešetřil (Q956806). Hell, Pavol; Nešetřil, Jaroslav (2004). Graphs and
Jaroslav_Nešetřil
Matrix in which most of the elements are zero
contrast, if most of the elements are non-zero, the matrix is considered dense. The number of zero-valued elements divided by the total number of elements
Sparse_matrix
Order-preserving mathematical function
The graph of a monotone operator G ( T ) {\displaystyle G(T)} is a monotone set. A monotone operator is said to be maximal monotone if its graph is a
Monotonic_function
On topological spaces where the intersection of countably many dense open sets is dense
topological space such that the intersection of countably many dense open sets is still dense). It is used in the proof of results in many areas of analysis
Baire_category_theorem
In graph theory, a strong orientation of an undirected graph is an assignment of a direction to each edge (an orientation) that makes it into a strongly
Strong_orientation
Python library for graphs and networks
tools for graph creation and analysis, producing visualizations of complex graphs can be challenging. Visualizing large or densely connected graphs may require
NetworkX
On short connecting nets with added points
for the dense instances of Steiner Tree problems. In a special case of the graph problem, the Steiner tree problem for quasi-bipartite graphs, S is required
Steiner_tree_problem
Graph coloring with equal color classes
that dense graphs are Hamiltonian: he conjectured that, if every vertex in an n-vertex graph has at least kn/(k + 1) neighbors, then the graph contains
Equitable_coloring
(small perturbations of a regular polygon) for which the β-skeleton is a dense graph with a quadratic number of edges. In the same quadratic time bound, the
Beta_skeleton
Dimensionality reduction of graph-based semantic data objects [machine learning task]
In representation learning, knowledge graph embedding (KGE), also called knowledge representation learning (KRL), or multi-relation learning, is a machine
Knowledge_graph_embedding
Therefore, other, less dense constraint graphs are considered. The horizontal visibility graph is a horizontal constraint graph in which the horizontal
Constraint_graph_(layout)
Conjugate transpose of an operator in infinite dimensions
been further extended to include unbounded densely defined operators, whose domain is topologically dense in, but not necessarily equal to, H . {\displaystyle
Hermitian_adjoint
Linear operator whose graph is closed
a closed graph. Hence, they cannot be defined on all of X {\displaystyle X} . To stay useful, they are instead defined on a proper but dense subspace
Closed_linear_operator
Contextual queries
people, or concepts relevant to the query. Tools like Google's Knowledge Graph provide structured relationships between entities to enrich query interpretation
Semantic_search
In graph theory, the mathematically simplest spatial network
In graph theory, a random geometric graph (RGG) is the mathematically simplest spatial network, namely an undirected graph constructed by randomly placing
Random_geometric_graph
Drawings of dense graphs have many crossings
plane drawing of a given graph, as a function of the number of edges and vertices of the graph. It states that, for graphs where the number e of edges
Crossing_number_inequality
travel, tourism, insurance
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travel, tourism, insurance