No involutions in the missing Moore graph

Abstract

The existence of a Moore graph of degree $57$ is the last open case in the Hoffman-Singleton classification of Moore graphs of diameter two. We prove that such a graph, if one exists, has no involutory automorphisms; consequently its automorphism group has odd order. The proof relies on a new trace-rank identity, valid for every finite graph and obtained from the Brauer quotient of $p$-permutation lattices: the trace of an automorphism of prime order $p$ on a spectral summand whose projection idempotent preserves $p$-adic permutation lattice equals the rank, over the residue field of characteristic $p$, of the corresponding idempotent restricted to the fixed vertices. For strongly regular graphs, this upgrades the classical character-value congruences for prime-order automorphisms to exact rank equalities. As further applications, the identity yields new restrictions on automorphisms of odd prime order of the hypothetical Moore graph, and it sharpens known automorphism analyses of other hypothetical strongly regular graphs.