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

Abstract

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 descendents has been done for the 5 classical families of graphs associated with short regular semilattices. We revisit and characterize these families in terms of posets consisting of descendents, and extend the classification to all of the 15 known infinite families with classical parameters and with unbounded diameter.

The Erdös-Ko-Rado theorem for twisted Grassmann graphs

“…The descendents 2 Leonard systems provide a linear algebraic framework characterizing the terminating branch of the Askey scheme [16] of (basic) hypergeometric orthogonal polynomials. ofJ q (2d + 1, d) have recently been classified by the author [24,Theorem 8.20]. To summarize:…”

“…In this setting, the "t-intersecting" condition amounts to requiring w d − t where d is the diameter of Γ, and we shall view the Erdős-Ko-Rado theorem as characterizing those subsets Y with w = d − t and w * = t by their sizes among all t-intersecting families. There are two steps involved: (1) construction of a specific feasible solution to the dual of a linear programming problem; (2) classification of the descendents [24] of Γ, i.e., those subsets having the property w + w * = d. We demonstrate this approach by deriving the Erdős-Ko-Rado theorem for the twisted Grassmann graphsJ q (2d + 1, d) discovered in 2005 by van Dam and Koolen [5]. the distance-i graph Γ i of Γ, so A 0 = I and d i=0 A i = J, the all ones matrix.…”

“…[17]), and that the induced subgraph is a Q-polynomial distance-regular graph provided it is connected; see [4,. See also [24] for more information on descendents. Now fix an integer t (0 < t < d) and suppose w d − t; in other words, Y is "t-intersecting".…”

“…Concerning the conclusion of Lemma 1, the classification of descendents has been done for the 15 known infinite families of Q-polynomial distance-regular graphs with so-called classical parameters and with unbounded diameter, including the above 4 families; see [4,23,24]. Moon [19] showed that the upper bound q d−t for H(d, q) and the characterization of its descendents as optimal intersecting families are valid under the (in general) weaker assumption q t + 2.…”

Bipartite Q-polynomial distance-regular graphs and uniform posets

“…In his thesis [12] Delsarte introduced the Q-polynomial property for a distance-regular graph Γ (see Section 2 for formal definitions). Since then the Q-polynomial property has been investigated by many authors, such as Bannai and Ito [1], Brouwer, Cohen and Neumaier [3], Caughman [4,5,6,7,8,9], Curtin [10,11], Jurišić, Terwilliger, andŽitnik [14], Lang [15,16], Lang and Terwilliger [17], Miklavič [18,19,20,21], Pascasio [22,23], Tanaka [24,25], Terwilliger [26,27,30,32], and Weng [33,34].…”