1. Introduction & Purposes

Expanders are special types of sparse yet highly connected graphs that play a fundamental rolein both theoretical and applied areas of computer science and mathematics. Their unique propertyof “expansion” means they have a low number of edges but still offer a high level of connectivity,making them efficient at spreading information or reaching all parts of the graph quickly. Thiscombination of sparse connectivity and robustness makes expanders extremely useful in designingefficient communication networks. They are also useful in algorithm designs, wheretheir structure can optimize processes like random sampling. Moreover, expanders canbe used to construct efficient error-correcting codes and cryptographic systems.

Pioneering work by Alon-Milman (1985), Dodziuk (1984), and Mohar (1989) demonstrated that the second-largest eigenvalue of the adjacency matrix of a regular graph serves as an effective measure of its expansion properties. Many explicit constructions of expander graphs arise from Cayley graphs, and the representation theory of finite groups provides powerful tools for analyzing their spectra. In this four-hours lecture series, we will give an expository overview of foundational and modern results in the theory of expander graphs, emphasizing the rich interplay between graph theory, linear algebra, group theory, and representation theory.

2. Outline & Descriptions

·       Whatare Expander Graphs?
Sparse yet highly connected graphs with strong expansion properties, useful in communication networks, algorithm design, error-correcting codes, and cryptography.

·       SpectralCharacterization of Expansion
The spectral gap (difference between largest and second-largest eigenvalues) is a key indicator of a graph’s expansion quality, as shown by the foundational work of Alon-Milman, Dodziuk, and Mohar.

·       Cayley Graphs and Representation Theory
Many expander constructions come from Cayley graphs. Representation theory of finite groups is a powerful tool for analyzing their spectra.

3. Prerequisites

Some foundational knowledge of Linear Algebra, Abstract Algebra, and Graph Theory is helpful.

4. Registration

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