Speaker: Sebastin Cioaba (University of Delaware)

Title: The smallest eigenvalues of Hamming, Johnson and other graphs

Abstract: The smallest eigenvalue of a graph is closely related to other graph parameters such as the independence number, the chromatic number or the max-cut. In this talk, I will describe the well-known connections between the smallest eigenvalue and the max-cut of a graph that have motivated various researchers such as Karloff, Alon, Sudakov, Van Dam, Sotirov to investigate the smallest eigenvalue of Hamming and Johnson graphs. The eigenvalues of the Hamming graphs are given by the Kravchuk (Krawtchouk) polynomials and the eigenvalues of the Johnson graphs are described by the Eberlein polynomials. I will describe our proofs of a conjecture by Van Dam and Sotirov on the smallest eigenvalue of (distance-j) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-j) Johnson graphs and mention some open problems. This is joint work with Andries Brouwer, Ferdinand Ihringer and Matt McGinnis.