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Sparse graph with strong connectivity
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander
Expander_graph
Spectral graph theory concept
Ramanujan graphs can be used to construct expander codes, which are good error correcting codes. Expander graph Alon-Boppana bound Expander mixing lemma
Ramanujan_graph
Graph defined from a mathematical group
The structure and symmetry of Cayley graphs make them particularly good candidates for constructing expander graphs. Let G {\displaystyle G} be a group
Cayley_graph
Topics referred to by the same term
turbine for high-pressure gas Expander graph, a sparse graph used in the combinatorics branch of mathematics StuffIt Expander, a computer file decompressor
Expander
theory, expander codes form a class of error-correcting codes that are constructed from bipartite expander graphs. Along with Justesen codes, expander codes
Expander_code
Measure of whether or not a graph has a "bottleneck"
of expander graphs as it is a way to measure the edge expansion of a graph. The so-called Cheeger inequalities relate the eigenvalue gap of a graph with
Cheeger constant (graph theory)
Cheeger_constant_(graph_theory)
Linear algebra aspects of graph theory
regular graph Algebraic connectivity Algebraic graph theory Spectral clustering Spectral shape analysis Estrada index Lovász theta Expander graph Weisstein
Spectral_graph_theory
Basic concept of graph theory
Cheeger constant (graph theory) Dynamic connectivity, Disjoint-set data structure Expander graph Strength of a graph Diestel, R. (2005). "Graph Theory, Electronic
Connectivity_(graph_theory)
instance, an even cycle is a cycle whose length is even. expander An expander graph is a graph whose edge expansion, vertex expansion, or spectral expansion
Glossary_of_graph_theory
The expander mixing lemma intuitively states that the edges of certain d {\displaystyle d} -regular graphs are evenly distributed throughout the graph. In
Expander_mixing_lemma
Graph nodes linked to, but not part of, a subgraph
particularly relevant for isoperimetric problems in graphs, separator theorems, minimum cuts, expander graphs, and percolation theory. Benjamini, Itai (2013)
Boundary_(graph_theory)
Concept in computing
when designing such protocols is that the neighbor set trace out an expander graph. Routing Tribler, BitTorrent peer-to-peer client using gossip protocol
Gossip_protocol
mathematical discipline of graph theory, the expander walk sampling theorem intuitively states that sampling vertices in an expander graph by doing relatively
Expander_walk_sampling
Geometric inequality applicable to any closed curve
are considered). In graph theory, isoperimetric inequalities are at the heart of the study of expander graphs, which are sparse graphs that have strong connectivity
Isoperimetric_inequality
American computer scientist
explicit constructions of constant-degree expanders of every size, starting from one constant-size expander. Crucial to the intuition and simple analysis
Salil_Vadhan
Time to reach all states of a Markov chain
least as large as this formula. Any n {\displaystyle n} -vertex regular expander graph also has expected cover time Θ ( n log n ) {\displaystyle \Theta (n\log
Cover_time
Type of knowledge base
knowledge graph is a knowledge base that uses a graph-structured data model or topology to represent and operate on data. Knowledge graphs are often used
Knowledge_graph
Archimedean solid with 62 faces
pentagrammic prisms. In the mathematical field of graph theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron
Rhombicosidodecahedron
Class of expander graphs arising in computational number theory
In mathematics, the supersingular isogeny graphs are a class of expander graphs that arise in computational number theory and have been applied in elliptic-curve
Supersingular_isogeny_graph
Computational hardness assumption
a certain class of expander graphs called "small set expanders" and other graphs that are very far from being small set expanders. This assumption implies
Small set expansion hypothesis
Small_set_expansion_hypothesis
Square matrix used to represent a graph or network
1016/0024-3795(68)90008-6. Shum, Kenneth; Blake, Ian (2003-12-18). "Expander graphs and codes". Volume 68 of DIMACS series in discrete mathematics and
Adjacency_matrix
Sex-specific adaptations
graph-theoretical parameters (e.g., minimum bipartition width, edge number, the expander graph property, minimum vertex cover), the structural connectome of women
Sexual_dimorphism
Topics referred to by the same term
Graph expansion may refer to: Expander graph Homeomorphism (graph theory) This disambiguation page lists articles associated with the title Graph expansion
Graph_expansion
Type of group in group theory
Zimmer can be used to construct expander graphs (Margulis), or even Ramanujan graphs (Lubotzky-Phillips-Sarnak). Such graphs are known to exist in abundance
Arithmetic_group
Partition of a graph's nodes into 2 disjoint subsets
Arora, Sanjeev; Rao, Satish; Vazirani, Umesh (2009), "Expander flows, geometric embeddings and graph partitioning", J. ACM, 56 (2), ACM: 1–37, doi:10.1145/1502793
Cut_(graph_theory)
Exactly solvable model of coupled oscillators
(2022). "Expander graphs are globally synchronising". arXiv:2210.12788 [math.CO]. Sloman, Leila (24 July 2023). "New Proof Shows That 'Expander' Graphs Synchronize"
Kuramoto_model
Method of graph decomposition
a graph, then the graph has treewidth k − 1. In particular, if k is the order of some bramble, then the treewidth is at least k − 1. Expander graphs of
Bramble_(graph_theory)
Second-largest eigenvalue lower bound
eigenvector. The graphs that come close to meeting this bound are Ramanujan graphs, which are examples of the best possible expander graphs. Its discoverers
Alon–Boppana_bound
Abstract devices built up of a fixed number of "wires"
by the Big-O notation. These are partly due to a construction of an expander graph. A simplified version of the AKS network was described by Paterson in
Sorting_network
Graph obeys some properties of random graphs
In graph theory, a graph is said to be a pseudorandom graph if it obeys certain properties that random graphs obey with high probability. There is no concrete
Pseudorandom_graph
used to disperse or dissolve pigments and other solids into a liquid. Expander graph Shaltiel, Ronen (2002). "Recent developments in explicit constructions
Disperser
Binary operation in graph theory
construction of constant-degree expander graphs. The construction is iterative, and needs as a basic building block a single, expander of constant size. In each
Zig-zag_product
Graph drawing with vertices in horizontal layers
Layered graph drawing or hierarchical graph drawing is a type of graph drawing in which the vertices of a directed graph are drawn in horizontal rows or
Layered_graph_drawing
Disperser Expander Extractor Bivariegated graph Cage (graph theory) Cayley graph Circle graph Clique graph Cograph Common graph Complement of a graph Complete
List_of_graph_theory_topics
Generalised concept of incidence structure of polygons
They are also related to expander graphs as they have nice expansion properties. Several classes of extremal expander graphs are obtained from generalized
Generalized_polygon
finite covolume. The theory of "group expansion" (expander graph properties of related Cayley graphs) for particular thin groups has been applied to arithmetic
Thin group (algebraic group theory)
Thin_group_(algebraic_group_theory)
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
American computer scientist and educator
developed the expander-flow method for graph partitioning. Their Journal of the ACM paper, "Expander flows, geometric embeddings and graph partitioning"
Satish_B._Rao
Israeli computer scientist and mathematician
zig-zag product of graphs, a method of combining smaller graphs to produce larger ones used in the construction of expander graphs. 2011: Elected as a
Avi_Wigderson
Graph representing social relations
social graph is a graph that represents social relations between entities. It is a model or representation of a social network. The social graph has been
Social_graph
2003 mathematics text
Mathematical Society Student Texts book series. In graph theory, expander graphs are undirected graphs with high connectivity: every small-enough subset
Elementary Number Theory, Group Theory and Ramanujan Graphs
Elementary_Number_Theory,_Group_Theory_and_Ramanujan_Graphs
Coding theory algorithm
typical class of Sipser–Spielman construction of expander codes, where the underlying graph is bipartite graph. Sipser and Spielman introduced a constructive
Zemor's_decoding_algorithm
Vertices connected in pairs by edges
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some
Graph_(discrete_mathematics)
Computer science algorithm
computer science, graph traversal (also known as graph search) refers to the process of visiting (checking and/or updating) each vertex in a graph. Such traversals
Graph_traversal
Cryptocurrency
The Graph is an open-source, decentralized protocol that powers the indexing and querying of blockchain data. It enables developers to build scalable
The_Graph
Israeli mathematician and former politician
applications of group theory to combinatorics and computer science (expander graphs) and error correcting codes. Lubotzky received the Erdős Prize in 1990
Alexander_Lubotzky
Study of mapping wiring diagrams
shows that the female connectome is better expander graph than the connectome of males. The better expanding property, the higher minimum bipartition width
Connectomics
American theoretical computer scientist (born 1954)
between expander graphs and derandomization. He and his PhD student Daniel Spielman introduced expander codes, an application of expander graphs. With fellow
Michael_Sipser
Theoretical computer scientist
Arora, Sanjeev; Rao, Satish; Vazirani, Umesh (2009). "Expander flows, geometric embeddings and graph partitioning". Journal of the ACM. 56 (2): 1–37. CiteSeerX 10
Sanjeev Arora (computer scientist)
Sanjeev_Arora_(computer_scientist)
Decomposition of a graph into hamiltonion cycles
In graph theory, a branch of mathematics, a Hamiltonian decomposition of a given graph is a partition of the edges of the graph into Hamiltonian cycles
Hamiltonian_decomposition
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
Russian mathematician
Margulis gave the first construction of expander graphs, which was later generalized in the theory of Ramanujan graphs. In 1986, Margulis gave a complete resolution
Grigory_Margulis
Unsolved problem in computational complexity theory
re-proved that unique games on expander graphs could be solved in polynomial time, and was one of (if not the) first graph algorithms to take advantage
Unique_games_conjecture
on a bounded-degree expander graph. Therefore, this logarithmic bound on the distortion of arbitrary graphs is tight. Planar graphs can be embedded with
GNRS_conjecture
Algorithm used for pathfinding and graph traversal
A* (pronounced "A-star") is a graph traversal and pathfinding algorithm that is used in many fields of computer science due to its completeness, optimality
A*_search_algorithm
Israeli computer scientist
randomly checking whether a few steps are true. Dinur creatively used expander graphs to prove the theorem via geometric means, and this connected the intuition
Irit_Dinur
Any planar graph can be subdivided by removing a few vertices
{\displaystyle 3.18{\sqrt {n}}} . Some sparse graphs do not have separators of sublinear size: in an expander graph, deleting up to a constant fraction of the
Planar_separator_theorem
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
Israeli mathematician and computer scientist
Johnson–Lindenstrauss lemma. Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (2006), "Expander graphs and their applications", Bulletin of the American Mathematical Society
Nati_Linial
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
that USTCON is in fact in L. This paper used expander graphs to guide the search through the input graph. Since USTCON is SL-complete, Reingold's result
SL_(complexity)
Israeli theoretical computer scientist
computer scientist whose research topics have included property testing, expander graphs, coding theory, and randomized algorithms with sublinear time complexity
Tali_Kaufman
Mathematics term
n ≥ 3) has property (T) to construct explicit families of expanding graphs, that is, graphs with the property that every subset has a uniformly large
Kazhdan's_property_(T)
Zig-zag product of graphs, a method of combining smaller graphs to produce larger ones used in the construction of expander graphs by Avi Wigderson. Development
List of Israeli inventions and discoveries
List_of_Israeli_inventions_and_discoveries
Mathematics reference book (2008)
on 99 specific mathematical concepts such as the axiom of choice, expander graphs, and Hilbert space. The second core section includes long surveys of
The Princeton Companion to Mathematics
The_Princeton_Companion_to_Mathematics
British mathematician
volume computation via random walks; finding edge disjoint paths in expander graphs, and exploring anti-Ramsey theory and the stability of routing algorithms
Alan_M._Frieze
Graph data structure
In computer science, an e-graph is a data structure that stores an equivalence relation over terms of some language. Let Σ {\displaystyle \Sigma } be
E-graph
Binary operation on mathematical graphs
an e-regular graph, then R is an (e + 1)-regular graph. Hoory, Shlomo; Linial, Nathan; Wigderson, Avi (7 August 2006). "Expander graphs and their applications"
Replacement_product
Mathematical concept
nilpotent. Other applications are to the construction of expander graphs from the Cayley graphs of finite simple groups, and to the related topic of superstrong
Approximate_group
Graph layout on multiple half-planes
distinction between these two complexity classes. The existence of expander graphs with constant page number is the key step in proving that there is
Book_embedding
Algorithm to search the nodes of a graph
tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores
Depth-first_search
Knowledge base to enhance search results
The Knowledge Graph is a knowledge base from which Google serves relevant information in an infobox beside its search results. This allows the user to
Knowledge_Graph_(Google)
Largest independent set of paired elements
The problem was formulated by Lawler (1976) as a common generalization of graph matching and matroid intersection. It is also known as polymatroid matching
Matroid_parity_problem
Algorithm that employs a degree of randomness as part of its logic or procedure
as the pairwise independence used in universal hashing the use of expander graphs (or dispersers in general) to amplify a limited amount of initial randomness
Randomized_algorithm
Israeli mathematician and computer scientist
problem of constructing a family of expander graphs with an explicit expansion ratio, useful in the design of fast graph algorithms. In 1995, Galil was inducted
Zvi_Galil
Procedures for constructing new graphs in graph theory
graph from an initial one by a complex change, such as: transpose graph; complement graph; line graph; graph minor; graph rewriting; power of graph;
Graph_operations
Heuristic search algorithm
science, beam search is a heuristic search algorithm that explores a graph by expanding the most promising node in a limited set. Beam search is a modification
Beam_search
Graph where every edge is in one triangle
In graph theory, a locally linear graph is an undirected graph in which every edge belongs to exactly one triangle. Equivalently, for each vertex of the
Locally_linear_graph
Type of sub-graph
recurrent and statistically significant subgraphs or patterns of a larger graph. All networks, including biological networks, social networks, technological
Network_motif
American mathematician
Davidoff is an American mathematician specializing in number theory and expander graphs. She is the Robert L. Rooke Professor of Mathematics and the chair
Giuliana_Davidoff
Graph width parameter
Arora, Sanjeev; Rao, Satish; Vazirani, Umesh (2009), "Expander flows, geometric embeddings and graph partitioning" (PDF), Journal of the ACM, 56 (2): A5:1–A5:37
Carving_width
Distributed computing model
the random walk induced by the scheduler and the graph's degree imbalance; on regular expander graphs this yields protocols matching the Θ(log n)-state
Population_protocol
Means to measure signal processing ability
(September 2009). "Efficient and Robust Compressed Sensing Using Optimized Expander Graphs" (PDF). IEEE Transactions on Information Theory. 55 (9): 4299–4308
Detection_theory
Graph database implemented in Java
global graph intelligence company that provides technology for analyzing and managing connected data. It is most known for creating the Neo4j Graph database
Neo4j
2002 science fiction novel by Australian author Greg Egan
in "Quantum Graph Theory", which holds that physical existence can be precisely modelled by complex constructions of mathematical graphs. However, the
Schild's_Ladder
Archimedean solid with 26 faces
("Archimedean solid") at MathWorld. Weisstein, Eric W. "Small rhombicuboctahedral graph". MathWorld. Klitzing, Richard. "3D convex uniform polyhedra x3o4x - sirco"
Rhombicuboctahedron
Israeli mathematician
theory to generalise the construction of Mikhail Gromov and others on expander graphs. His fundamental work will have major implications even outside mathematics
Shai_Evra
Mathematics prize
Shlomo Hoory, Nathan Linial, and Avi Wigderson for their article "Expander graphs and their applications". 2007: Jeffrey Weeks for his article "The Poincaré
Levi_L._Conant_Prize
Electronic calculator capable of plotting graphs
A graphing calculator (also graphics calculator or graphic display calculator) is a handheld computer that is capable of plotting graphs, solving simultaneous
Graphing_calculator
American physicist
Complexes: a generalization of classical statistical mechanics on expander graphs". Quantum Information and Computation. 14 (1&2): 144–180. arXiv:1301
Matthew_Hastings
Complements of perfect graphs are perfect
In graph theory, the perfect graph theorem of László Lovász (1972a, 1972b) states that an undirected graph is perfect if and only if its complement graph
Perfect_graph_theorem
Hungarian mathematician (born 1982)
Mathematics and Mathematical Statistics. Varjú studied the construction of expander graphs with number-theoretic methods involving arithmetic groups[A] and questions
Péter_Varjú
zig-zag product to efficiently construct expander graphs. Jesper Jansson. Deterministic Space-Bounded Graph Connectivity Algorithms. Manuscript. 1998
Symmetric_Turing_machine
Theorem in computational complexity theory
discovered a significantly simpler proof of the PCP theorem, using expander graphs. She received the 2019 Gödel Prize for this. A version of the PCP theorem
PCP_theorem
Graph with at most one crossing per edge
In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing
1-planar_graph
In computer science, polymorphic recursion (also referred to as Milner–Mycroft typability or the Milner–Mycroft calculus) refers to a recursive parametrically
Polymorphic_recursion
American mathematician
completing her Ph.D. in 2006. Her doctoral dissertation, Pseudocodewords, Expander Graphs and the Algebraic Construction of Low-Density Parity-Check Codes, was
Christine_Kelley
Solid with six equal square faces
drawing a graph with vertices connected with an edge in a plane. Such a graph is called the cubical graph, a special case of the hypercube graph. The cube
Cube
chain. It can be seen as a special case of Cheeger inequalities in expander graphs. Let X {\displaystyle X} be a finite set and let K ( x , y ) {\displaystyle
Cheeger_bound
Graph of intervisible locations in computational geometry
visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane. Each node in the graph represents
Visibility_graph
EXPANDER GRAPH
EXPANDER GRAPH
Male
Dutch
, defender of man.
Boy/Male
Muslim
Servant of the expander, Extender
Girl/Female
Arabic, Australian, Muslim
Vastness; Expanse
Boy/Male
Arabic, Muslim
Servant of the Expander (Allah)
Boy/Male
Hindu, Indian, Marathi
To Expand
Boy/Male
Latin Scottish
Fought with Aeneas.
Boy/Male
Indian
Servant of the expander, Extender
Boy/Male
Arabic, Muslim
Servant of the Expander
Boy/Male
Muslim
The expander
Boy/Male
Arabic
Creator; Servant of the Expander
Boy/Male
American, Arabic, French, German, Greek, Gujarati, Indian, Kannada, Muslim
Splendid; Defender of Mankind
Boy/Male
Gujarati, Hindu, Indian, Kannada, Sanskrit
To Expand; Progress
Boy/Male
Shakespearean
Pericles, Prince of Tyre' A Pander.
Boy/Male
Muslim/Islamic
Servant of the Expander
Boy/Male
Arabic
Expounder of Islamic Law
Boy/Male
Arabic, Muslim
Servant of the Expander (Allah)
Boy/Male
Arabic
Expander; Spreader; One who Enlarges
Boy/Male
Indian
The expander
Girl/Female
Indian, Sanskrit
Awakened; Roused; Expanded
Boy/Male
Muslim
Expounder of Islamic Law.
EXPANDER GRAPH
EXPANDER GRAPH
Boy/Male
Muslim
Worthy of description
Boy/Male
Hindu, Indian, Telugu
Name of Vishnu
Boy/Male
African, American, Anglo, Australian, British, Christian, English
From the Town by the Lake
Boy/Male
Tamil
Lord Shiva
Boy/Male
Australian, Iranian, Parsi
A Character in Shahnameh
Boy/Male
Greek American
People's victory.
Girl/Female
Arabic, Muslim
Present; Gift; Singular of Nihel
Girl/Female
Australian, Chinese, Danish, Finnish, German, Hebrew, Italian, Latin, Swedish
Graced with God's Bounty; God is Gracious; God has Shown Favour; Derived from a Compound of Anna and Liesa
Surname or Lastname
English and Scottish
English and Scottish : variant of Copeland.
Girl/Female
Muslim/Islamic
The light e.g. nurul islam, the light of islam
EXPANDER GRAPH
EXPANDER GRAPH
EXPANDER GRAPH
EXPANDER GRAPH
EXPANDER GRAPH
imp. & p. p.
of Expand
n.
That which is expanded; expanse; extend surface; as, the expansion of a sheet or of a lake; the expansion was formed of metal.
imp. & p. p.
of Pander
v. t.
To expand.
a.
Unfolded; expanded; exhibited conspicuously or ostentatiously.
a.
Extensive; expanded.
p. pr. & vb. n.
of Pander
v. i.
To become expanded or inflated; to swell.
v. t.
To cause the particles or parts of to spread themselves or stand apart, thus increasing bulk without addition of substance; to make to occupy more space; to dilate; to distend; to extend every way; to enlarge; -- opposed to contract; as, to expand the chest; heat expands all bodies; to expand the sphere of benevolence.
a.
That expands, or may be expanded; extending; spreading; enlarging.
v. t.
To lay open by extending; to open wide; to spread out; to diffuse; as, a flower expands its leaves.
n.
Anything which causes expansion esp. (Mech.) a tool for stretching open or expanding a tube, etc.
v. i.
To act the part of a pander.
n.
An indefinite expanse.
n.
That which is expanded or spread out; a wide extent of space or body; especially, the arch of the sky.
v. t.
To play the pander for.
a.
Expanded; enlarged.
v. i.
To become widely opened, spread apart, dilated, distended, or enlarged; as, flowers expand in the spring; metals expand by heat; the heart expands with joy.
n.
Expanse.
v. t.
To state in enlarged form; to develop; as, to expand an equation. See Expansion, 5.