Multiplicity formulas for full bipartite dual-polar graph eigenvalues

Determine the multiplicities of the eigenvalues of the adjacency matrix of the full bipartite graph of a non-bipartite dual polar graph with respect to a fixed vertex, including the multiplicity of every eigenvalue displayed in Theorem 7.8.

Background

The paper proves that the full bipartite graph of a non-bipartite dual polar graph is Q-polynomial and explicitly determines the eigenvalues of its adjacency matrix in Theorem 7.8. However, the corresponding eigenvalue multiplicities are not computed. The problem is therefore to complete the spectral description of these graphs by determining those multiplicities.

The authors note that, by Corollary 7.7, it is enough to compute the multiplicity of an irreducible Terwilliger-algebra module with endpoint r and diameter d for every 0 ≤ r ≤ D and 0 ≤ d ≤ D. They indicate that Theorem 2.5(3) of Terwilliger’s work [13] may be useful for this computation.