Codes and Expansions (CodEx) Seminar

Zilin Jiang (Arizona State University)
Forbidden subgraphs and spherical two-distance sets

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 \((1-\alpha)/(\beta-\alpha) < 2.0198\). Joint work with Alexandr Polyanskii, Jonathan Tidor, Yuan Yao, Shengtong Zhang and Yufei Zhao.