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In combinatorics, an expander graph refers to a sparse graph which has high connectivity properties, quantified using vertex or edge expansion as described below. Expander constructions have spawned research in pure and applied mathematics, with several applications to computer science, and in particular to theoretical computer science, design of robust computer networks and the theory of error-correcting codes. Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ...
In the mathematical subfield of numerical analysis a sparse matrix is a matrix populated primarily with zeros. ...
In mathematics and computer science the connectivity of graphs is one of the basic concepts of graph theory. ...
In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ...
This article just presents the basic definitions. ...
Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...
Computer science (informally, CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ...
A computer network is a system for communication between computers. ...
In information theory and coding, an error-correcting code or ECC is a code in which each data signal conforms to specific rules of construction so that departures from this construction in the received signal can generally be automatically detected and corrected. ...
Definitions
There are several different ways to measure the expansion properties of a finite, undirected multigraph G. A multigraph is a graph with multiple edges, i. ...
Edge expansion The edge expansion h(G) of a graph G is defined as  where the minimum is over all nonempty sets S of at most n / 2 vertices, and  stands for the set of edges with exactly one endpoint in S.
Vertex expansion The α-vertex expansion gα(G) of a graph G is defined as  where Γ(S) stands for the set of vertices with at least one neighbor in S. In a variant of this definition (called unique neighbor expansion) Γ(S) stands for the set of vertices in V with exactly one neighbor in S.
Spectral expansion When G is regular, a linear algebraic definition of expansion is possible based on the eigenvalues of the adjacency matrix A = A(G) of G (where Aii is the number of loops at the ith vertex). Because A is symmetric, the Spectral theorem implies that A has n real-valued eigenvalues  . Because G is regular, λ0 = d where d is the degree of regularity of G. In some contexts, the spectral gap of G is defined to be d − λ1. In other contexts, the spectral gap refers to d − λ, where  . In graph theory, a regular graph is a graph where each vertex has the same number of neighbors, i. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ...
In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In mathematics and computer science, the adjacency matrix for a finite graph on n vertices is an n × n matrix where the nondiagonal entry aij is the number of edges joining vertex and vertex , and the diagonal entry aii is either twice the number of loops at vertex or just...
In linear algebra, a symmetric matrix is a matrix that is its own transpose. ...
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...
In graph theory, the degree (or valency) of a vertex is the number of edges incident to the vertex. ...
The normalized version of this definition, where we consider the matrix A / d and get eigenvalues between -1 and 1, is also widely used and more convenient in the statement of some results.
Frequently one will refer to the second-largest eigenvalue of a graph G, which refers to the max of  and  . Depending on the context it may be either the normalized version (taking value between 0 and 1) or the un-normalized version (taking value between 0 and d).
Expander families A family  of d-regular graphs is an edge expander family if there is a constant c > 0 such that  for each  . Typically we want d to be constant, though it is sometimes also interesting to consider d = log | G | or even d = | G | O(1). Similarly,  is a vertex expander family if there is a constant c > 1 such that  for each , and is a spectral expander family if some positive constant is a lower bound for the spectral gap of each .
These definitions can be extended to the case of directed graphs. A directed graph can also be interpreted as a balanced bipartite graph (with all edges going from one copy of V to another copy). The definition of bipartite expanders can further be generalized to the case of unbalanced bipartite graphs. This article just presents the basic definitions. ...
In the mathematical field of graph theory, a bipartite graph is a special graph where the set of vertices can be divided into two disjoint sets with two vertices of the same set never sharing an edge. ...
Relationship between the different definitions The expansion parameters defined above are related to each other. In particular, for any graph G,  . Consequently, every vertex expander family is also an edge expander family.
Similarly, when G is d-regular, there is a relationship between h(G) and the spectral gap d − λ1 of G. An inequality due to "Cheeger and Buser in the continuous case and Tanner, Alon, and Milman in the discrete case" [1] states that
 As a result, a family of graphs is an edge expander family if and only if is a spectral expander family.
Examples of expanders A random d-regular graph has good expansion, with high probability. Ramanujan graphs are a family of d-regular expander graphs with d being constant and with explicit constructions, that have essentially the largest possible spectral gap. Algebraic constructions based on Cayley graphs are known for various variants of expander graphs. Most recently, combinatorial constructions of expanders have also been discovered. // About Bees This article is about completely random and illogical things. ...
In combinatorics, an expander graph is in rough terms a sparse graph with high vertex or edge expansion, or in other words highly connected. ...
Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
The Cayley graph of the free group on two generators a and b In mathematics, a Cayley graph, named after Arthur Cayley, is a graph that encodes the structure of a group. ...
Applications and Useful Properties The original motivation for expanders is to build economical robust networks (phone or computer): an expander with bounded valence is precisely an asymptotic robust graph with number of edges growing linearly with size (number of vertices), for all subsets.
Expander graphs have found extensive applications in computer science, in designing algorithms, error correcting codes, extractors, pseudorandom generators, sorting networks and robust computer networks. They have also been used in proving many important results in computational complexity theory, such as SL=L and the PCP Theorem. In Cryptography too, expander graphs are used to construct hash functions. Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...
Flowcharts are often used to represent algorithms. ...
In information theory and coding, an error-correcting code or ECC is a code in which each data signal conforms to specific rules of construction so that departures from this construction in the received signal can generally be automatically detected and corrected. ...
Mathematics An (N,M,D,K,e)-extractor is a bipartite graph with N nodes on the left and M nodes on the right such that each node on the left has D neighbors (on the right), which has the added property that for any subset A of N of...
Let G be a deterministic polynomial time function from N<ω to N<ω with stretch function l:N → N, so that if x has length n then G(x) has length l(n). ...
A sorting network is a circuit with n input connections for data items to be presented in an arbitrary order, and n output connections on which the items are to be output in sorted order. ...
A computer network is a system for communication between computers. ...
In computer science, computational complexity theory is the branch of the theory of computation that studies the resources, or cost, of the computation required to solve a given problem. ...
In computational complexity theory, SL (Symmetric Logspace) is the complexity class of problems log-space reducible to USTCON, which is the problem of determining whether there exists a path between two vertices in an undirected graph, otherwise described as the problem of determining whether two vertices are in the same...
In computational complexity theory, L is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a logarithmic amount of memory space. ...
Overview The PCP Theorem is a theorem in Computational complexity theory Theorem . ...
The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages. ...
A hash function (or hash algorithm) is a way of creating a small digital fingerprint from any kind of data. ...
The following are some properties of expander graphs that have proven useful in many areas.
Expander Mixing Lemma -
The expander mixing lemma states that, for any two subsets of the vertices  , the number of edges between S and T is approximately what you would expect in a random d-regular graph, i.e.  . The expander mixing lemma states that, for any two subsets of an expander graph , the number of edges between and is approximately what you would expect in a random d-regular graph, i. ...
Expander Walk Sampling -
The Chernoff bound states that, when sampling many independent samples from a random variables in the range [ − 1,1], with high probability the average of our samples is close to the expectation of the random variable. The expander walk sampling lemma, due to Gillman and Ajtai-Komlós-Szemerédi, states that this also holds true when sampling from a walk on an expander graph. This is particularly useful in the theory of derandomization, since sampling according to an expander walk uses much fewer random bits than sampling independently. The expander walk sampling theorem, the earlist version of which is due to Ajtai-Komlós-Szemerédi and the more general version typically attributed to Gillman, states that sampling from an expander graph is almost as good as sampling independently. ...
In probability theory, Chernoffs inequality, named after Herman Chernoff, states the following. ...
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