No involutions in the missing Moore graph

Abstract

The Moore graph of degree $57$, if one exists, is the remaining open case of the Hoffman-Singleton classification in diameter two. Although its existence remains open, substantial restrictions on the automorphism group of such a graph are known. In this paper we prove that a Moore graph of degree $57$ has no involutory automorphisms. The proof combines the known fixed-point structure of an involution with a module-theoretic obstruction. More precisely, we consider the vertex module over the ring of 2-adic integers and the direct summand given by the image of the spectral idempotent for the eigenvalue $-8$. Comparing the ordinary trace of the involution on this summand with the dimension of its Brauer quotient gives a contradiction.