Expander graphs a lso form a basis o f the work b y alon et al. In fact one important aspect of asymptotic group theory the part of group theory concerned with studying the. Note that these notions are the direct analogues of the standard notions of vertex and edge expansion in graphs. We say a graph g is c,dsemiregular if it is bipartite with vertex partition s,t, such that all vertices in s have degree c, and all vertices in t have degree d,withc network coding, and expander graphs ho yee cheung, lap chi lau, kai man leung the chinese university of hong kong abstract we present a new algebraic formulation to compute edge connectivities in a directed graph, using the ideas developed in network coding. We will now see that graphs satisfying this condition are very good approximations of the complete graph. This dissertation is brought to you for free and open access by the dissertations at tigerprints. In particular, we generate 50 scalefree networks 1 and 50 random. We will typically interpret the properties of expander graphs in an asymptotic sense. First, we present an introduction to the line of work that began with the study of expander graphs in the nonconstructive setting, which then led to the algebraic con. Each of these is a wonderful source of information, but we. Luca trevisan c 2011, 20, 2014 by luca trevisan this work is licensed under the creative commons attributionnoncommercialnoderivs 3. Luca trevisan c 2011, 20, 2014 by luca trevisan this work is. Gonality of expander graphs 3 on the graph with positive rank.
A very brief overview of expander graphs, which were the topic of my honours thesis. An expander code is a code whose factor graph is an expander graph. It is therefore related to the books of lubotzky 74 and his colloquium lectures 75, of sarnak 96, and of tao 104, and to the detailed survey of hoory, linial and wigderson 51. An introduction to expander graphs people eth zurich. Pdf expander graphs are highly connected sparse finite graphs. Heres a list of some of the things that expander graphs can be used to do. Well often abuse nomenclature slightly, and just refer to the expander g j, or even just g, omitting. Pdf cryptographic hash functions from expander graphs. However in order to do so it is useful to have some understanding of the analogous spectral notions of amenability and kazhdan property t which are.
Since all these vertices are distinct, the choice made at. Expander graphs are one of the deepest tools of theoretical computer science and discrete mathematics, popping up in all sorts of contexts since their introduction in the 1970s. Lecture notes on expansion, sparsest cut, and spectral. The algorithmic problem of finding a small balanced cut in a graph that is, of finding a certificate that a graph is not an. A computer science application of expander graphs 11 acknowledgments 14 references 14 1. In mathematics, the supersingular isogeny graphs are a class of expander graphs that arise in computational number theory and have been applied in ellipticcurve cryptography. They play an important role in computer science as basic building blocks for network constructions. The divisorial gonality of a graph is always less than or equal to the tree gonality, since the pullback of a vertex via a harmonic morphism has positive rank. In this lecture, we will focus on expander graphs also called expanders, which are pseudorandom objects in a more restricted sense than what we saw in the last two lectures. Lecture notes on expansion, sparsest cut, and spectral graph. Layouts of expander graphs a a b b a b c a b figure 1.
Graph connectivities, network coding, and expander graphs. An introduction to expander graphs emmanuel kowalski download. Kolmogorov and barzdins proof on the realization of networks in threedimensional space 4 3. We note that it is now more standard to use the bipartite version of expander graphs due to zemor zem01. We recall the notions of basic graph theory and we note that we will work with undirected, loop free multiplicity free. Expander graphs a ubiquitous pseudorandom structure professor. The complete graph has the best expansion property, but it has largest possible degree. An expander code where the factor graph is a n,m,d. Expander graphs a very brief introduction gilesgardam.
Pdf expander graphs in pure and applied mathematics. An introduction to expander graphs emmanuel kowalski. We study the gaps in the set of the eigenvalues of the adjacency matrices of cubic graphs, with special attention to large gaps. However, for us, it is slightly easier notationally to use the. The goal of this book is to give an introduction to expander graphs and their applications. These two insights allow us to give a different perspective on an existing construction rsw06, show that standard iterative constructions of expanders can be adjusted to give expander towers almost for free. E cient robust secret sharing from expander graphs brett hemenway1 and rafail ostrovsky2. We are not interested in the expander parameter of a single graph gbut an entire family of graphs fg ig. Informally, a graph is a good expander if it has low. Expander graphs and the zigzag product 1 introduction informally, an expander is an undirected graph that has relatively sparse density, but whose vertices are nevertheless highly connected.
These are graphs that are sparse yet very wellconnected. May 24, 2005 expander graphs are one of the deepest tools of theoretical computer science and discrete mathematics, popping up in all sorts of contexts since their introduction in the 1970s. A related invariant is the tree gonality, which is the smallest degree of a harmonic morphism from the graph to a tree. In this paper, we investigate other such maximally gapped intervals and sets, and. Here we construct two families of a,bregular graphs that expand both locally and globally.
However in order to do so it is useful to have some understanding of the analogous spectral notions of amenability and kazhdan property t which are important for in nite groups. We recall the notions of basic graph theory and we note that we will work with undirected, loop free multiplicity free graphs. This property can be viewed from several di erent angles. Family of circulant graphs and its expander properties. S is an edge expander and an jsjtriple expander for some 0. While random regular graphs are expanders with high probability, they almost surely fail to expand locally.
Expander graphs expander graphs are highly connected sparse graphs. Request pdf expander graphs and their application a major consideration we had in. Consequently, expanders have the property that any small subset of the vertices has a large set of neighbors outside of the set. While random regular graphs are expanders with high probability, they almost surely fail to.
The mathematics of expander graphs is studied by three distinct communities. Jacob fox 1 expander graphs expander graphs are graphs with the special property that any set of vertices s unless very large has a number of outgoing. Expander graphs are highly connected sparse finite graphs. There is a constant dsuch that for every n, there is a dregular graph on nvertices. Converting a a monotone matching to b a book embedding and c a queue layout 21, 15, 38. Next we use the probabilistic method to show that such graphs do exist. Expansion and expander graphs informally, a graph is a good expander if it is simultaneously sparse and highly connected. Pdf gap sets for the spectra of cubic graphs semantic. Expander graphs found numerous applications in the theory of computation, ranging from the design of sorting networks 1 to the proof that undirected connectivity is decidable in determinstic logspace. It has been accepted for inclusion in all dissertations by an authorized administrator of tigerprints. In one aspect, an expander graph is walked to compute a hash function. First, we present an introduction to the line of work that began with the study of expander. The expander graph is walked using respective subsets of an input message.
Well often abuse nomenclature slightly, and just refer to the expander g j, or even just g, omitting explicit mention of the entire family of graphs. Additionally their construction results in lossless expander graphs with constant degree d. Expander graphs, the subject of these notes, are certain families of graphs, becoming larger and larger, which have the following two competing properties. Call step i a coincidence if it is free and moreover if the randomly selected. A label of a last vertex walked is an output of the hash function. Dylan has already covered most of the background which we will need with the exception to the idea of bipartite expander graphs called magical graphs.
Cryptographic hash functions from expander graphs article pdf available in journal of cryptology 221. Expander graphs and their application request pdf researchgate. A disconnected graph is not an expander, since the boundary of a connected component is empty. We say that a dregular graph on nvertices is an expander if for every subset sof at most n2 vertices, j sj 54jsj. Hash function constructions from expander graphs are described. Expander graphs now that we have seen a variety of basic derandomization techniques, we will move on to study the. Expander graphs and their applications shlomo hoory, nathan linial, and avi wigderson an overview 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. The discovery of kolmogorov and barzdin in this section, we will introduce the concept of expander graphs and attempt. We are not interested in the expander parameter of a. Threshold secret sharing is a protocol that allows a dealer to share a secret among npl. Us7382876b2 hash function constructions from expander.
Application of expander graphs to errorcorrecting codes. Expander graphs and their applications shlomo hoory, nathan linial, and avi wigderson an overview a major consideration we had in writing this survey was to make it. Graph connectivities, network coding, and expander graphs ho yee cheung, lap chi lau, kai man leung the chinese university of hong kong abstract we present a new algebraic formulation to compute. Expander graphs found numerous applications in the theory of computation, ranging from the design of sorting networks 1 to the proof that undirected connectivity is decidable in determinstic logspace 15.
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