It is known that the expansion property of a graph influences the performance of the corresponding code when decoded using iterative algorithms. Certain graph products may be used to obtain larger expander graphs from smaller ones. In particular, the zig-zag product and replacement product may be used to construct infinite families of constant degree expander graphs. This paper investigates the...

In this work, we present the first local-decoding algorithm for expander codes. This yields a new family of constant-rate codes that can recover from a constant fraction of errors in the codeword symbols, and where any symbol of the codeword can be recovered with high probability by reading N symbols from the corrupted codeword, where N is the block-length of the code. Expander codes, introduce...

The design of LDPC codes based on a class of expander graphs is investigated. Graph products, such as the zig-zag product [9], of smaller expander graphs have been shown to yield larger expanders. LDPC codes are designed based on the zigzag product graph of two component Cayley graphs. The results for specific cases simulated reveal that the resulting LDPC codes compare well with other random L...

We present a construction of expander graphs obtained from Cayley graphs of narrow ray class groups, whose eigenvalue bounds follow from the Generalized Riemann Hypothesis. Our result implies that the Cayley graph of (Z/qZ)∗ with respect to small prime generators is an expander. As another application, we show that the graph of small prime degree isogenies between ordinary elliptic curves achie...

In this paper, we consider a structural and geometric property of graphs, namely the presence large expanders. The problem finding such structures was first considered by Krivelevich [SIAM J. Disc. Math. 32 1 (2018)]. Here, show that induced subgraph is an expander can be reduced to simpler topological minor expander. Our proof constructive, which helpful in algorithmic setting.
 We also e...

Four different ways of obtaining low-density parity-check codes from expander graphs are considered. For each case, lower bounds on the minimum stopping set size and the minimum pseudocodeword weight of expander (LDPC) codes are derived. These bounds are compared with the known eigenvalue-based lower bounds on the minimum distance of expander codes. Furthermore, Tanner’s parity-oriented eigenva...

In this note we determine exactly the expansion rate of an infinite 4-regular expander graph which is a variant of an expander due to Margulis. The vertex set of this graph consists of all points in the plane. The point (x, y) is adjacent to the points S(x, y), S(x, y), T (x, y), T(x, y) where S(x, y) = (x, x+y) and T (x, y) = (x+y, y). We show that the expansion rate of this 4-regular graph is...

We show how to compute O( √ logn)-approximations to Sparsest Cut and Balanced Separator problems in Õ(n) time, thus improving upon the recent algorithm of Arora, Rao and Vazirani (2004). Their algorithm uses semidefinite programming and required Õ(n) time. Our algorithm relies on efficiently finding expander flows in the graph and does not solve semidefinite programs. The existence of expander ...

This paper is dedicated to present a proof of the Spectral Theorem, and to discuss how the Spectral Theorem is applied in combinatorics and graph theory. In this paper, we also give insights into the ways in which this theorem unveils some mysteries in graph theory, such as expander graphs and graph coloring.

In this paper, we study the large-scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a graph is robustly expanding if it still expands after the deletion of a small fraction of its vertices and edges. Our main result allows us to harness the us...