It is shown that the edges of any n-point vertex expander can be replaced by new edges so that the resulting graph is an edge expander, and such that any two vertices that are joined by a new edge are at distance O( √ log n) in the original graph. This result is optimal, and is shown to have various geometric consequences. In particular, it is used to obtain an alternative perspective on the re...

We introduce and study analogues of expander hyperfinite graph sequences in the context directed acyclic graphs, which we call "extender" "hypershallow" sequences, respectively. Our main result is a probabilistic construction non-hypershallow sequences.

Our construction of error-correcting codes will exploit bipartite expander graphs (as these give a much cleaner construction than the general case). Let’s begin by examining what a bipartite expander graph should look like. It’s vertex set will have two parts, U and V , each having n vertices. Every vertex will have degree d, and every edge will go from a vertex in U to a vertex in V . In the s...

1 Introductory graph and metric notions 5 1.1 The Cheeger constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Expander graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Isoperimetric dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Rough isome...

There are many excellent existing texts for the material in this lecture, starting with Lubotzky's monograph [11] and recent AMS survey paper [10]. For expander graphs and their use in theoretical computer science, check the survey by Hoory, Linial and Wigderson [6]. We give here a brief introduction. We start with a definition. Definition 0.1. (Expander graph) A finite connected k-regular grap...

In recent years a high dimensional theory of expanders has emerged. The notion of combinatorial expanders of graphs (i.e. the Cheeger constant of a graph) has seen two generalizations to high dimensional simplicial complexes. One generalization, known as coboundary expansion is due to Linial and Meshulem; the other, which we term here systolic expansion, is due to Gromov, who showed that systol...

Within the METIS-II project, we have implemented a machine translation system which uses transfer and expander rules to build an AND/OR graph of partial translation hypotheses and a statistical ranker to find the best path through the graph. The paper gives an overview of the architecture and an evaluation of the system for several languages.

6 Spectral Graph Theory 16 6.1 Graph Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6.2 Polynomial interlacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.2.1 An Application to Expander Graphs . . . . . . . . . . . . . . . . . . 18 6.3 Connectivity as a Minimization Problem . . . . . . . . . . . . . . . . . . . . 20 6.4 Courant-Fisher and Alg...

Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any bounded-degree nvertex graph, the union of two random spanning trees approximates the expansion of every cut of the graph to within a factor of O(log n). For the random graph Gn,p, for p = Ω(log n/n), we give a randomized algor...

We show how expander based arguments may be used to prove that message passing algorithms can correct a linear number of erroneous messages. The implication of this result is that when the block length is sufficiently large, once a message passing algorithm has corrected a sufficiently large fraction of the errors, it will eventually correct all errors. This result is then combined with known r...