Vertex Subsets with Minimal Width and Dual Width in $Q$-Polynomial Distance-Regular Graphs

Vertex Subsets with Minimal Width and Dual Width in Q-Polynomial Distance-Regular Graphs

We study Q-polynomial distance-regular graphs from the point of view of what we call descendents, that is to say, those vertex subsets with the property that the width w and dual width w∗ satisfy w+w∗ = d, where d is the diameter of the graph. We show among other results that a nontrivial descendent with w > 2 is convex precisely when the graph has classical parameters. The classification of de...

Width and dual width of subsets in polynomial association schemes

The width of a subset C of the vertices of a distance-regular graph is the maximum distance which occurs between elements of C. Dually, the dual width of a subset in a cometric association scheme is the index of the “last” eigenspace in the Q-polynomial ordering to which the characteristic vector of C is not orthogonal. Elementary bounds are derived on these two new parameters. We show that any...

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...

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.

Q-polynomial distance-regular graphs with a1=0