Distance-regular graphs where the distance-d graph has fewer distinct eigenvalues

Let the Kneser graph K of a distance-regular graph Γ be the graph on the same vertex set as Γ, where two vertices are adjacent when they have maximal distance in Γ. We study the situation where the Bose-Mesner algebra of Γ is not generated by the adjacency matrix of K. Let Γ be a distance-regular graph of diameter d on n vertices. Let Γi be the graph with the same vertex set as Γ where two vertices are adjacent when they have distance i in Γ. Let A be the adjacency matrix of Γ, and Ai that of Γi. We are interested in the situation where Ad has fewer distinct eigenvalues than A. In this situation the matrix Ad generates a proper subalgebra of the Bose-Mesner algebra of Γ, a situation reminiscent of imprimitivity. We survey the known examples, derive parameter conditions, and obtain strong results in what we called the ‘half antipodal’ case. Unexplained notation is as in [BCN]. The vertex set X of Γ carries an association scheme with d classes, where the i-th relation is that of having graph distance i (0 ≤ i ≤ d). All elements of the Bose-Mesner algebra A of this scheme are polynomials of degree at most d in the matrix A. In particular, Ai is a polynomial in A of degree i (0 ≤ i ≤ d). Let A have minimal idempotents Ei (0 ≤ i ≤ d). The column spaces of the Ei are common eigenspaces of all matrices in A. Let Pij be the corresponding eigenvalue of Aj , so that AjEi = PijEi (0 ≤ i, j ≤ d). Now A has eigenvalues θi = Pi1 with multiplicities mi = rkEi = trEi (0 ≤ i ≤ d). Index the eigenvalues such that θ0 > θ1 > · · · > θd. Standard facts about Sturm sequences give information on the sign pattern of the matrix P . Proposition 1 Let Γ be distance-regular, and P its eigenvalue matrix. Then row i and column i of P both have i sign changes. In particular, row d and column d consist of nonzero numbers that alternate in sign. 2 1Research supported by the Ministerio de Ciencia e Innovación, Spain, and the European Regional Development Fund under project MTM2011-28800-C02-01, and the Catalan Research Council under project 2009SGR1387.