# 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.