Symbolic proof and higher-diameter extensions of the threshold method

Determine whether the finite intersection-parameter exclusion used for the spectral threshold admits a substantially shorter symbolic proof, and determine how far the same threshold method can be extended to generalized odd graphs of larger diameter.

Background

The paper proves its diameter-three classification by combining analytic valency bounds with an exact computer-assisted enumeration of feasible intersection arrays. The finite search reduces the valency to at most 182 and then excludes all but the arrays of C_7, the Odd graph O_4, and the folded 7-cube.

The authors identify two unresolved methodological directions: replacing the computational exclusion with a substantially shorter symbolic argument, and extending the threshold approach to larger diameter. The latter is potentially difficult because the number of intersection parameters increases with the diameter, although the paper's finite-reduction proposition applies to any fixed positive target.