In this work and its sequel, we study the expanding phenomenon of matrices over a finite chain ring large residue field. A sum-product estimate is proved. It shown that [Formula: see text] moderate expander on with exponent text]. These results generalize main theorems in recent [29] Xie Ge. The proofs use spectral graph theory elementary divisor theory.

The main contribution of this work is a new type of graph product, which we call the zig-zag product. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of arbitrary size, startin...

Consider the undirected graph Gn = (Vn, En) where Vn = (Z/nZ) and En contains an edge from (x, y) to (x+ 1, y), (x, y + 1), (x+ y, y), and (x, y + x) for every (x, y) ∈ Vn. Gabber and Galil, following Margulis, gave an elementary proof that {Gn} forms an expander family. In this expository note, we present a somewhat simpler proof of this fact, and demonstrate its utility by isolating a key pro...

Recent work has attempted to identify structure in social and information graphs by using the following approach: first, use random walk methods to explore the neighborhood of a node; second, use ideas from natural language processing to use this neighborhood information to learn vector representations of these nodes reflecting properties of the graph. Informally, the idea is that if a node is ...

Graph-based codes were introduced in the previous lecture. Definitions as well as important properties of those codes were presented with explicit decoding algorithms left over to this lecture.As mentioned previously, one of the main topics in the near future would be linear time algorithms. A linear time decoding algorithm of the Expander Codes will be discussed in the following part. As the e...

Expanders are classes of highly connected graphs that are of fundamental importance in graph theory, with numerous applications, especially in theoretical computer science [29]. While the literature contains various definitions of expanders, this paper focuses on bipartite expanders. For ∈ (0, 1], a bipartite graph G with bipartition V (G) = A ∪ B is a bipartite -expander if |A| = |B| and |N(S)...

Up until the Bourgain-Gamburd 2005 breakthrough the only known ways to turn SLd(Fp) into an expander graph (i.e. to find a generating set of small size whose associated Cayley graph has a good spectral gap) was either through property (T ) (as in the Margulis construction) when d > 3 or through the Selberg property (and the dictionary between combinatorial expansion of the Cayley graphs and the...

Linear expanders have numerous applications to theoretical computer science. Here we show that a regular bipartite graph is an expander ifandonly if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result, which has an analytic analogue for Riemannian manifolds enables one to generate expanders randomly and check efficiently their expanding properties...

We give a simple recipe for translating walks on Cayley graphs of a group G into a quantum operation on any irrep of G. Most properties of the classical walk carry over to the quantum operation: degree becomes the number of Kraus operators, the spectral gap becomes the gap of the quantum operation (viewed as a linear map on density matrices), and the quantum operation is efficient whenever the ...