# Bipartite Q-polynomial distance-regular graphs and uniform posets

@article{Miklavic2011BipartiteQD, title={Bipartite Q-polynomial distance-regular graphs and uniform posets}, author={Stefko Miklavic and Paul M. Terwilliger}, journal={Journal of Algebraic Combinatorics}, year={2011}, volume={38}, pages={225-242} }

Let Γ denote a bipartite distance-regular graph with vertex set X and diameter D≥3. Fix x∈X and let L (resp., R) denote the corresponding lowering (resp., raising) matrix. We show that each Q-polynomial structure for Γ yields a certain linear dependency among RL2, LRL, L2R, L. Define a partial order ≤ on X as follows. For y,z∈X let y≤z whenever ∂(x,y)+∂(y,z)=∂(x,z), where ∂ denotes path-length distance. We determine whether the above linear dependency gives this poset a uniform or strongly…

## 13 Citations

### The Folded (2D + 1)-cube and Its Uniform Posets

- Mathematics
- 2018

Let Γ denote the folded (2D + 1)-cube with vertex set X and diameter D ≥ 3. Fix x ∈ X. We first define a partial order ≤ on X as follows. For y, z ∈ X let y ≤ z whenever ∂(x, y) + ∂(y, z) = ∂(x, z).…

### The quantum adjacency algebra and subconstituent algebra of a graph

- MathematicsJ. Comb. Theory, Ser. A
- 2019

### Distance-regular graphs

- Mathematics
- 2014

An introduction to distance-regular graphs is presented for the reader who is unfamiliar with the subject, and an overview of some developments in the area of distance- regular graphs since the monograph 'BCN' was written.

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

- MathematicsArs Mathematica Contemporanea
- 2019

In this paper we consider a distance-regular graph $\Gamma$. Fix a vertex $x$ of $\Gamma$ and consider the corresponding subconstituent algebra $T$. The algebra $T$ is the $\mathbb{C}$-algebra…

### Uniform posets and Leonard pairs based on unitary spaces over finite fields

- Mathematics
- 2016

Let be the -dimensional unitary space over finite field and let denote the orbit of subspaces of under the unitary group. Denote by the set of subspaces generated by . By ordering by ordinary…

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

- Mathematics
- 2022

This survey paper contains a tutorial introduction to distance-regular graphs, with an emphasis on the subconstituent algebra and the Q-polynomial property.

### The Attenuated Space Poset $\mathcal{A}_q(N, M)$

- Mathematics
- 2016

In this paper, we study the incidence algebra $T$ of the attenuated space poset $\mathcal{A}_q(N, M)$. We consider the following topics. We consider some generators of $T$: the raising matrix $R$,…

### Compatibility and companions for Leonard pairs

- MathematicsThe Electronic Journal of Linear Algebra
- 2022

In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let $\mathbb{F}$ denote a field, and let $V$ denote a…

## References

SHOWING 1-10 OF 33 REFERENCES

### The Parameters of Bipartite Q-polynomial Distance-Regular Graphs

- Mathematics
- 2002

Let Γ denote a bipartite distance-regular graph with diameter D ≥ 3 and valency k ≥ 3. Suppose θ0, θ1, ..., θD is a Q-polynomial ordering of the eigenvalues of Γ. This sequence is known to satisfy…

### Bipartite Q-Polynomial Distance-Regular Graphs

- MathematicsGraphs Comb.
- 2004

It is shown that C is Q-polynomial if and only if one of the following holds: C is the ordinary 2D-cycle, which is the Hamming cube HðD, or the antipodal quotient of Hð2D; 2Þ.

### Tails of Bipartite Distance-regular Graphs

- MathematicsEur. J. Comb.
- 2002

Abstract Let Γ denote a bipartite distance-regular graph with diameterD ≥ 4 and valency k ≥ 3. Let θ 0 > θ 1 > ⋯ > θD denote the eigenvalues of Γ and let E0, E1,⋯ , EDdenote the associated…

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

- MathematicsElectron. J. Comb.
- 2011

It is shown among other results that a nontrivial descendent with $w\ge 2$ is convex precisely when the graph has classical parameters.

### On the Multiplicities of the Primitive Idempotents of a Q-Polynomial Distance-regular Graph

- MathematicsEur. J. Comb.
- 2002

By proving the above theorem, Ito, Tanabe and Terwilliger's notion of a tridiagonal pair is resolved and a conjecture of Dennis Stanton is resolved.

### The Terwilliger Algebra of a 2-Homogeneous Bipartite Distance-Regular Graph

- MathematicsJ. Comb. Theory, Ser. B
- 2001

This work describes the simple T-modules, a 2-homogeneous bipartite distance-regular graph with diameter D?3 and valency k?3, and gives three sets of generators for T, two of which satisfy the relations of the quantum universal enveloping algebra of the Lie algebra sl(2).