# Codes and Expansions (CodEx) Seminar

## Zilin Jiang (Arizona State University)Spherical two-distance sets and spectral theory of signed graphs

A set of unit vectors in a Euclidean space is called a spherical two-distance set if the pairwise inner products of these vectors assume only two values $\alpha > \beta$. It is known that the maximum size of a spherical two-distance grows quadratically as the dimension of the Euclidean space grows. However when the values $\alpha$ and $\beta$ are held fixed, a very intricate behavior of the maximum size emerges. Building on our recent resolution in the equiangular case, that is $\alpha + \beta = 0$, we make a plausible conjecture which connects this behavior with spectral theory of signed graphs in the regime $\beta < 0 < \alpha$, and we confirm this conjecture when $\alpha + 2\beta < 0$ or $(1-\alpha)/(\beta - \alpha)$ is $1$, $\sqrt{2}$ or $\sqrt{3}$.

Joint work with Jonathan Tidor, Yuan Yao, Shengtong Zhang and Yufei Zhao.