Ramanujan Graphs and Interlacing Families

TL;DR

This survey explains the development and applications of interlacing families and Ramanujan graphs over the past decade, highlighting their significance in spectral graph theory and connections to other mathematical areas.

Contribution

It provides a comprehensive overview of the interlacing families method and its role in constructing bipartite Ramanujan graphs of all degrees, emphasizing recent advances and connections.

Findings

Explains the interlacing families method and its applications.

Highlights the construction of bipartite Ramanujan graphs of all degrees.

Connects the method to various areas of mathematics.

Abstract

This survey accompanies a lecture on the paper ``Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees'' by A. Marcus, D. Spielman, and N. Srivastava at the 2024 International Congress of Basic Science (ICBS) in July, 2024. Its purpose is to explain the developments surrounding this work over the past ten or so years, with an emphasis on connections to other areas of mathematics. Earlier surveys about the interlacing families method by the same authors focused on applications in functional analysis, whereas the focus here is on applications in spectral graph theory.