Informally, an expander is an undirected graph that has relatively sparse density, but whose vertices are nevertheless highly connected. Consequently, expanders have the property that any small subset of the vertices has a large set of neighbors outside of the set. This simple graph property has led to highly useful results in a number of branches of computer science. In addition, the study of ...

We initiate the study of approximating the largest induced expander in a given graph G. Given a ∆-regular graph G with n vertices, the goal is to find the set with the largest induced expansion of size at least δ ·n. We design a bi-criteria approximation algorithm for this problem; if the optimum has induced spectral expansion λ our algorithm returns a λ log δ exp(∆/λ) -(spectral) expander of s...

The expander constructions based on algebraic methods can give expanders that are both explicit (i.e. we can quickly construct the graph, or even obtain neighborhood information without constructing the entire graph, and Ramanujan, meaning that the spectral gap is essentially as large as possible. It also follows from this spectral bound that the edge expansion of Ramanujan graphs is essentiall...

In this lecture, we shall look at Expander Codes, Tanner Codes which are a generalization of Expander Codes and Distance amplification codes. These codes visualize the code as a graph where the value of the ith bit of a codeword corresponds to, for example, the value associated with the ith vertex or the ith edge of the graph. The properties of the graph determine the properties such as distanc...

A locally testable code is a code defined by a robust set of local constraints. Namely, the distance of a vector from the code is well approximated by the fraction of local constraints that it violates. A constraint graph of an LTC is a graph whose vertices are labeled by the coordinates of the code, and vertex i is adjacent to j whenever they occur together in a constraint. We study the relati...

Expander graphs are known to facilitate effective routing and most real-world networks have expansion properties. At the other extreme, it has been shown that in some special graphs, removing certain edges can lead to more efficient routing. This phenomenon is known as Braess’s paradox and is usually regarded as a rare event. In contrast to what one might expect, we show that Braess’s paradox i...

In this lecture we study the explicit constructions of expander graphs. Although we can construct expanders probabilistically, this does not suffice for many applications. • One applications of expander graphs is for reducing the randomness complexity of algorithms (cmp. Lecture 6), thus constructing the graph itself randomly does not serve this purpose. • Sometimes we may even need expanders o...

Graph expansion has proved to be a powerful general tool for analyzing the behavior of routing algorithms and the inter{connection networks on which they run. We develop new routing algorithms and structural results for bounded{degree expander graphs. Our results are uniied by the fact that they are all based upon, and extend, a body of work asserting that expanders are rich in short, disjoint ...