Expander Graphs and Random Walks in Computer Science

Expander graphs are foundational structures in theoretical computer science, cryptography, and network design, offering dense connectivity with remarkably sparse edges. This course provides a clear, accessible introduction to both the combinatorial and algebraic views of expanders and explores how random walks on these graphs power modern derandomization techniques. You will build a rigorous conceptual foundation starting with core graph definitions and spectral properties, progressing into practical mathematical tools like the Expander Mixing Lemma and Chernoff-type concentration bounds for random walks. What you'll learn: - Understand the fundamental definitions of combinatorial and spectral expander graphs - Analyze graph spectra, adjacency eigenvalues, and the spectral gap - Apply the Expander Mixing Lemma to quantify edge distribution and connectivity - Evaluate random walks on expanders and compute set-hitting probabilities - Explore expander-based Chernoff bounds and their role in randomness reduction - Practice theoretical proofs and structural calculations through written exercises The course begins by establishing key terminology, graph theory basics, and linear algebraic concepts before progressing into spectral expansion theorems and probability bounds. It is designed for learners, students, and engineers looking to understand expander graphs without needing prior advanced knowledge in spectral graph theory. Start reading today to master the mathematical principles behind expander graphs and pseudorandomness.