Summer 2023 - MATH 895 G100

Reading (4)

Class Number: 2199

Delivery Method: In Person


  • Course Times + Location:

    Location: TBA



Topic: Expansion, Lifts, and Interlacing Families
This is an ambitious course aimed at understanding one of the great recent developments
in combinatorics, namely the Marcus-Spielman-Srivastava interlacing polynomial method and its application to prove the existence of bipartite Ramanujan graphs of all possible degrees.
In preparation for reading this paper we will take approximately 3 weeks to introduce each of the following three important topics in algebraic graph theory:
(1) The spectrum of the adjacency matrix and the matching polynomial. Spectrum of the adjacency matrix: structural connections and Cayley graphs. Matching polynomial: real rootedness and absolute bounds.
(2) Graph expansion, including connections to the second largest eigenvalue, mixing of random walks, the Alon-Boppana bound, and random constructions.
(3) Lifts of graphs and random lifts, including relationships to the spectrum, graph structure.
Finally, with this material in hand we will delve into the (first) masterpiece of Marcus-SpielmanSrivastava: Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees. This paper is beautifully written and can be approached without much background, but the preceding investigation will establish the basic results on which this paper is based, and give the appropriate context to understand the main result.

Prerequisite: Graph theory and exposure to algebraic graph theory.


  • Assignments 60%
  • Final Project 40%


Course Structure and Grades: We will meet twice per week with the class beginning much like a standard class with lectures covering basic material. Later in the course students will take turns presenting material with each student giving at least two presentations during the course. There will be four (large) homework assignments given out during the course which students will be permitted to work on in groups. Each assignment will be worth 15% of the final grade. The remaining 40% of the grade will be based upon a final project.



References: We will use a variety of sources for this course. Spectral Graph Theory by Godsil and Royle will be a key resource for the first part. An online course given by Linial and Wigderson on graph expansion will be a key resource for the second part. The third part will call upon papers of Linial and Lov´asz. The final part will be the main paper for the course by Marcus, Spielman, and Srivastava.


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