Ramanujan Graphs and Interlacing Families

2024-12-30

AI Summary

This paper resolves the long-standing open problem of explicitly constructing bipartite Ramanujan graphs: it provides, for the first time, a deterministic, polynomial-time construction of infinite families of bipartite Ramanujan graphs for **all integer degrees** $d geq 3$. Methodologically, it pioneers and systematically applies the *interlacing families* framework—integrated with determinant polynomial analysis, random signed matrices, and Lax-type spectral perturbation theory—to reduce spectral gap estimation to analyzing root distributions of associated polynomials. The key contribution is a rigorous proof of the existence of infinite families of bipartite Ramanujan graphs whose second-largest eigenvalue magnitude achieves the optimal bound $O(sqrt{d})$, thereby refuting the Karpovsky–Miller conjecture. This work establishes a profound connection between spectral graph theory and algebraic polynomial theory, yielding novel tools for coding theory, explicit expander construction, and combinatorial design in quantum computing.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Deterministic Planning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Computational social science

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.

Problem

• Bipartite Ramanujan Graphs
• Interlacing Families
• Graph Theory

Innovation

• Interlacing Families
• Ramanujan Graphs
• Bipartite Graphs