On Q-polynomial distance-regular graphs Γ with strongly regular graphs Γ2 and Γ3

On bipartite Q-polynomial distance-regular graphs

Let Γ denote a bipartite Q-polynomial distance-regular graph with vertex set X, diameter d ≥ 3 and valency k ≥ 3. Let RX denote the vector space over R consisting of column vectors with entries in R and rows indexed by X. For z ∈ X, let ẑ denote the vector in RX with a 1 in the z-coordinate, and 0 in all other coordinates. Fix x, y ∈ X such that ∂(x, y) = 2, where ∂ denotes path-length distance...

Distance-Regular Graphs with Strongly Regular Subconstituents

In [3] Cameron et al. classified strongly regular graphs with strongly regular subconstituents. Here we prove a theorem which implies that distance-regular graphs with strongly regular subconstituents are precisely the Taylor graphs and graphs with a1 = 0 and ai ∈ {0, 1} for i = 2, . . . , d .

Q-polynomial distance-regular graphs with a1=0

Almost-bipartite distance-regular graphs with the Q-polynomial property

Let Γ denote a Q-polynomial distance-regular graph with diameter D ≥ 4. Assume that the intersection numbers of Γ satisfy ai = 0 for 0 ≤ i ≤ D − 1 and aD 6= 0. We show that Γ is a polygon, a folded cube, or an Odd graph.

Nonsymmetric Askey-Wilson polynomials and Q-polynomial distance-regular graphs

In his famous theorem (1982), Douglas Leonard characterized the q-Racah polynomials and their relatives in the Askey scheme from the duality property of Q-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or Laurent) version of the q-Racah polynomials in the above situation. Let Γ denote a Q-polynomial distance-regular graph that contains a Delsarte clique C. Assume ...