Expander graphs and their applications

Expander Graphs and Their Applications

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields. But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathema...

CSL 863 : Expander graphs and their applications

Clustering is the task of grouping vertices of a graph which are more tightly knit than others. In the age of the internet and large social networks, these networks tend to be prohibitively large to run algorithms of time complexity larger than Õ(n) . In some cases, like the internet for example, it would be impossible to access all of the nodes even once. This calls for more near-linear and lo...

PCPs and Inapproximability: Expander Graphs and their Applications

Expander Graphs

This paper will introduce expander graphs. Kolmogorov and Barzdin’s proof on the three dimensional realization of networks will be discussed as one of the first examples of expander graphs. The last section will discuss error correcting code as an application of expander graphs to computer science.

Expander Graphs

Let AG be the adjacency matrix of G. Let λ1 ≥ λ2 ≥ . . . ≥ λn be the eigenvalues of AG. Sometimes we will also be interested in the Laplacian matrix of G. This is defined to be LG = D−AG, where D is the diagonal matrix where Dvv equals the degree of the vertex v. For d-regular graphs, LG = dI −AG, and hence the eigenvalues of LG are d− λ1, d− λ2, . . . , d− λn. Lemma 1. • λ1 = d. • λ2 = λ3 = . ...