Spectral gaps in groups and graphs

From CanisiusmathWiki

Ryan Grady, Samuel Cooper, Dominic Dotterrer, Stratos Prassidis, and Clara Brasseur in the computer room.

Work based on this project

Talks given at the Young Mathematicians Conference at Ohio State University

• Kazhdan's Property (T) for Graphs by Clara E Brasseur
• Covers, Laplacians, and Heat Kernels on Directed Graphs by Ryan E Grady
• Calculating the Ihara Zeta Function for Some Infinite Graphs by Samuel D Cooper
• Spectral Properties of the Zig-Zag Product of Graphs by Dominic Dotterrer

Papers located online at arXiv.org

1. math.CO/0607352 [1] (http://www.arxiv.org/pdf/math.CO/0607352) :

   Title: Properties of the Generalized Zig-Zag Product of Graphs
   Authors: Samuel Cooper, Dominic Dotterrer, Stratos Prassidis
   Comments: 15 pages
   Subj-class: Combinatorics
   MSC-class: 05C50
    

2. math.CO/0607351 [2] (http://www.arxiv.org/pdf/math.CO/0607351) :

   Title: Kazhdan's Property (T) for Graphs
   Authors: Clara Brasseur, Ryan E. Grady, Stratos Prassidis
   Comments: 13 pages
   Subj-class: Combinatorics
   MSC-class: 05C25

Papers located at the Canisius College website

There are links to the following two preprints from REU 2005 at [3] (http://www3.canisius.edu/~prasside/)

Some Properties of Zig-Zag Products of Graphs (joint work with Brian Meagher, David Mittiga).

Laplacians, Heat Kernels and Kesten's Theorem (joint work with Tricia Profic, Jack Wessell).

Information related to this project

Kazhdan Graph Structures

• A Definition for Kazhdan Graph Structures

Generalized Zig-zag Products of Graphs

• The Definition of the Generalized Zig-Zag Product
• Lemmas for Zig-zag products of simple graphs
• Constructing Covers of Zig-Zag Products
• Degenerate H-labellings and Bipartite Cloud Splittings
• Edge Considerate Labellings and Cloud Cluster connections
• Vertex Considerate Labellings
• Eigenvalues of the Zig-zag Product for Restricted Labellings

Theorem 3.1

• An example of the complications of Matrix factorization when H is not regular
• The method for bounding the second term does not hold without the regularity of G
• Matrix factorization in Theorem 3.1 in: "Some Properties Of Zig-Zag Products Of Graphs" by Meagher, Mittiga, and Prassidis
• A more explicit explanation of the decomposition of x in the proof of Theorem 3.1 (cited above)

Covers, Laplacians, and Heat Kernels of Directed Graphs

• The Definition of a Combinatorial Cover of a Directed Graph
• The Spectrum of a Directed Graph and Its Combinatorial Cover
• Cited Publications

Zeta Functions of Graph Bundles

• Definition and Basic Properties of Graph Bundles

Reference and Related Comments

• An explanation (and a possible generalization) of the rotational map defined in: "The Evolution of Expander Graphs" by David Xiao
• Doubly Stochastic Matrices and their application in Theorem 3.1