An Explicit Construction of Expander Graphs
ثبت نشده
چکیده
2 The Construction Fix n = m2 for a natural m and let An =Zm Zm,Zm being the group of integers modulo m. An may be thought of a combinatorial torus. Consider the following 5 bijections on An: 1. σ0 : (x;y) 7! (x;y), 2. σ1 : (x;y) 7! (x;x+ y), 3. σ2 : (x;y) 7! (x;x+ y+1), 4. σ3 : (x;y) 7! (x+ y;y), and 5. σ4 : (x;y) 7! (x+ y+1;y), addition modulo m. Now define Gn = (Un;Vn; En) as follows: Un = Vn = An, and En = f(u;σ(u)) : u 2Un;σ2 fσigg. Observe that, as defined, Gn is a multigraph. Our goal is to prove the following theorem:
منابع مشابه
An Explicit Construction of an Expander Family
This paper proves that there exist infinite families of graphs which satisfy a uniform lower bound on their spectral gap. We first prove existence via a probabilistic method. The explicit construction involves various results from algebra and representation theory, which we explore at some length. It turns out that the expander family we construct achieves a maximal condition on expander famili...
متن کاملHighly Symmetric Expanders
Expander graphs are relevant to theoretical computer science in addition to the construction of high-performance switching networks. In communication network applications, a high degree of symmetry in the underlying topology is often advantageous, as it may reduce the complexity of designing and analyzing switching and routing algorithms. We give explicit constructions of expander graphs that a...
متن کاملEfficient Compressed Sensing using Lossless Expander Graphs with Fast Bilateral Quantum Recovery Algorithm
Compressed Sensing is a novel approach to bypass the Nyquist sampling limits whenever the signals are sparse, and to compress them simultaneously. In this paper, improving our previous results, we will propose a compressed sensing algorithm based on the high-quality lossless unbalanced vertex expander graphs, with a fast and simple quantum decoding algorithm. Exploiting the unique neighborhood ...
متن کاملSymmetric Groups and Expanders
We construct an explicit generating sets Fn and F̃n of the alternating and the symmetric groups, which make the Cayley graphs C(Alt(n), Fn) and C(Sym(n), F̃n) a family of bounded degree expanders for all sufficiently large n. These expanders have many applications in the theory of random walks on groups and other areas of mathematics. A finite graph Γ is called an ǫ-expander for some ǫ ∈ (0, 1), ...
متن کاملExplicit expanders with cutoff phenomena
The cutoff phenomenon describes a sharp transition in the convergence of an ergodic finite Markov chain to equilibrium. Of particular interest is understanding this convergence for the simple random walk on a bounded-degree expander graph. The first example of a family of bounded-degree graphs where the random walk exhibits cutoff in total-variation was provided only very recently, when the aut...
متن کاملThe zig-zag product
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...
متن کامل