A Computer-Assisted Proof of the Uniqueness of the Perkel Graph

Abstract

The Perkel graph is a distance-regular graph of order 57, degree 6 and diameter 3, with intersection array (6, 5, 2; 1, 1, 3). We describe a computer assisted proof that every graph Γ with this intersection array is isomorphic to the Perkel graph. The computer proof relies heavily on the fact that the minimal idempotents for Γ, and their submatrices, are positive semidefinite.

To minimize the risk of computer errors we have used two different methods to establish the same theorem and as an added precaution large parts of the corresponding programs were written by different authors.

The first method generates plausible subgraphs induced by all vertices at distance 3 from a fixed vertex of Γ and then tries to extend each of the generated graphs to a full graph with the given intersection array.

The second method generates possible neighborhoods for a pentagon in Γ. It turns out that every such pentagon can be extended to a Petersen graph in Γ. We then prove mathematically that there is, up to isomorphism, only a single graph Γ with this property.