On the second eigenvalue and linear expansion of regular graphs.

Abstract. The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the best-known explicit expanders. The spectral method yielded a lower bound of k\4 on the expansion of Iinear-sized subsets of k-regular Ramanujan graphs. We improve the lower bound ontheexpansion of Ramanujan graphs to approximately k/2, Moreover. we construct afamilyof k-regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k/2. This shows that k/2 is the best bound one can obtain using the second eigenwdue method. We also show an upper bound of roughly 1 +~on the average degree of linear-sized induced subgraphs of Ramanujan graphs. This compares positively with the classical bound 2~. As a byproduct, we obtain improved results on random walks on expanders and construct selection networks (respectively, extrovert graphs) of smaller size (respectively, degree) than was previously known

Expander Graphs and Coding Theory

Expander graphs are highly connected sparse graphs which lie at the interface of many different fields of study. For example, they play important roles in prime sieves, cryptography, compressive sensing, metric embedding, and coding theory to name a few. This thesis focuses on the connections between sparse graphs and coding theory. It is a major challenge to explicitly construct sparse graphs with good expansion properties, for example Ramanujan graphs. Nevertheless, explicit constructions do exist, and in this thesis, we survey many of these constructions up to this point including a new construction which slightly improves on an earlier edge expansion bound. The edge expansion of a graph is crucial in applications, and it is well-known that computing the edge expansion of an arbitrary graph is NP-hard. We present a simple algo-rithm for approximating the edge expansion of a graph using linear programming techniques. While Andersen and Lang (2008) proved similar results, our analysis attacks the problem from a different vantage point and was discovered independently. The main contribution in the thesis is a new result in fast decoding for expander codes. Current algorithms in the literature can decode a constant fraction of errors in linear time but require that the underlying graphs have vertex expansion at least 1/2. We present a fast decoding algorithm that can decode a constant fraction of errors in linear time given any vertex expansion (even if it is much smaller than 1/2) by using a stronger local code, and the fraction of errors corrected almost doubles that of Viderman (2013)

Computationally efficient error-correcting codes and holographic proofs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1995.Includes bibliographical references (p. 139-145).by Daniel Alan Spielman.Ph.D