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

Abstract

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

Distance-regular graphs, the subconstituent algebra, and the $Q$-polynomial property

“…Comprehensive treatments can be found in [3,32,36,37,[67][68][69]78]. In addition, there are papers about the thin condition [10,21,24,64,69,82], irreducible T -modules with endpoint one [30], Γ being bipartite [6,13,14,41], Γ being almost-bipartite [9,42], Γ being dual bipartite [22], Γ being almost dual bipartite [23], Γ being 2-homogeneous [15,17,18,53], Γ being tight [56], Γ being a hypercube [27], Γ being a Doob graph [63], Γ being a Johnson graph [49,62], Γ being a Grassmann graph [48], Γ being a dual polar graph [84], Γ having a spin model in the Bose-Mesner algebra [16,52]. Some miscellaneous topics about irreducible T -modules can be found in [26,34,35,40,43,44,55,59,60,…”

Distance-regular graphs

“…The Q-polynomial generalized odd graphs have been classified by Lang and Terwilliger [438]: the folded (2D+1)-cube, the Odd graph O D+1 , and the graphs with D = 3 satisfying…”

A diagram associated with the subconstituent algebra of a distance-regular graph

“…So it is natural to compute the irreducible T -modules. These modules are important in the study of hypercubes [14,26], dual polar graphs [20,38], spin models [6,10], codes [13,28], the bipartite property [3,4,9,16,21,22,23,25,27], the almost-bipartite property [5,8,17], the Q-polynomial property [3,7,11,12,18,19,27,33], and the thin property [15,24,30,31,32,34,36,37].…”