Explicit Concentrators from Generalized N-Gons

Eigenvalues and expanders

“…For example we show that a regular graph G is a strong expander if and only if 2(G) is well separated from 0. The "if" part is not too difficult and is somewhat similar to a result of Tanner [30]. The "only if" part is much trickier and is the discrete analogue of a theorem of Cheeger [12] on Riemannian manifolds.…”

Eigenvalues in Combinatorial Optimization

“…By the work of Tanner [129] and later by several other authors (cf. [96]) it became clear that λ 2 measures the connectivity in the following sense: How difficult is to split the graph into two large pieces?…”

The Minimum Distance of Graph Codes

“…The graph may be seen as a simple expander graph: The eigenvalues are ±(q + 1) and ± √ q (all real since A is symmetric). (See [4].) Starting from a vertex in the right set, q + 1 vertices in the left set can be reached in one transition, and q(q + 1) vertices in the right set can be reached from these vertices.…”

“…Certain bipartite graphs derived from generalized polygons have good expansion properties [4], and hence the codes derived from these have large minimum distances. The generalized polygons are incidence structures consisting of points and lines where any point is incident with the same number of lines, and any line is incident with the same number of points.…”

“…In 1981 Tanner [1] introduced a construction of error-correcting codes based on graphs and since then a considerable number of results have been obtained [2], [3], [4], [5] and [6]. The recent textbook by Roth [8] contains a thorough presentation of the subject.…”