A Characterization of Q-Polynomial Distance-Regular Graphs Using the Intersection Numbers

Abstract

We consider a primitive distance-regular graph Γ with diameter at least 3. We use the intersection numbers of Γ to find a positive semidefinite matrix G with integer entries. We show that G has determinant zero if and only if Γ is Q-polynomial.

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

“…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 [26,34,35,40,43,44,55,59,60,75,83].…”