Nonexistence of certain pseudogeometric graphs

Abstract

Izo($r$) is the pseudogeometric graph for pG${}_r(s,t)$ which satisfies the Krein equality and of which the local graph also satisfies the Krein equality. We prove that Izo($r$) exists if and only if a spherical tight 4-design on $S^{{(2r+1)}^2-4}$ exists. Nonexistence of infinitely many spherical tight 4-designs is well known. It gives the nonexistence of Izo($r$) for infinitely many $r$. Especially, Izo(4) does not exist, which was the first open case.