What is edge expansion?

The edge expansion of a graph G=(V,E) is the minimal edge expansion over all of its vertex subsets whose size is at most |V|/2. Bounded degree trees have small edge expansion. In this figure, we consider the circled subset S of 4 vertices in a 4regular graph.

What is called Graph Theory?

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines).

Why are expander graphs important?

Expander graphs have found extensive applications in computer science, in designing algorithms, error correcting codes, extractors, pseudorandom generators, sorting networks (Ajtai, Komlós & Szemerédi (1983)) and robust computer networks.

What is expansion on a graph?

The edge expansion of a subset of vertices S ⊆ V in a graph. G measures the fraction of edges that leave S. In a d-regular. graph, the edge expansion/conductance Φ(S) of a subset. S ⊆ V is defined as Φ(S) = |E(S,V \S)|

Which is the best definition of an expander graph?

Intuitively, an expander is a finite, undirected multigraph in which every subset of the vertices that is not “too large” has a “large” boundary. Different formalisations of these notions give rise to different notions of expanders: edge expanders, vertex expanders, and spectral expanders, as defined below.

What is the formal definition of an expander?

The formal definition of the most basic expander is as follows: Let k≥2 be an integer, and let X be a k-regular graph (that is, each vertex v∈V has exactly kneighbors). The Cheeger constant h of Xis defined to be h(X)= min ∅⊂F⊂V = = |∂F| min(|F|,|V\\F|) where ∂Fis the set of edges running from Fto its complement V\\Fand |∂F|is its cardinality.

Can a disconnected graph be an expander graph?

A disconnected graph is not an expander, since the boundary of a connected component is empty. Every connected graph is an expander; however, different connected graphs have different expansion parameters.

What are the applications of Expander constructions in mathematics?

Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes.