Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and let A denote the adjacency matrix of Γ. For x∈X and for 0≤i≤D, let Γi(x) ...denote the set of vertices in X that are distance i from vertex x. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. We first show that Δ2=0 implies that D≤5 or c2∈{1,2}.
For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A,E0⁎,E1⁎,…,ED⁎, where for 0≤i≤D, Ei⁎ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T-module W we mean min{i|Ei⁎W≠0}.
In this paper we assume Γ has the property that for 2≤i≤D−1, there exist complex scalars αi, βi such that for all x,y,z∈X with ∂(x,y)=2, ∂(x,z)=i, ∂(y,z)=i, we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|. We additionally assume that Δ2=0 with c2=1.
Under the above assumptions we study the algebra T. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2. We give an orthogonal basis for this T-module, and we give the action of A on this basis.
Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and let A denote the adjacency matrix of Γ. For x∈X and for 0≤i≤D, let Γi(x) ...denote the set of vertices in X that are distance i from vertex x. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. It is known that Δ2=0 implies that D≤5 or c2∈{1,2}.
For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A,E0∗,E1∗,…,ED∗, where for 0≤i≤D, Ei∗ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T-module W we mean min{i|Ei∗W≠0}.
We find the structure of irreducible T-modules of endpoint 2 for graphs Γ which have the property that for 2≤i≤D−1, there exist complex scalars αi, βi such that for all x,y,z∈X with ∂(x,y)=2,∂(x,z)=i,∂(y,z)=i, we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|, in case when Δ2=0 and c2=2. The case when Δ2=0 and c2=1 is already studied by MacLean et al. 15.
We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2 and it is not thin. We give a basis for this T-module, and we give the action of A on this basis.
The notion of multivariate P- and Q-polynomial association scheme has been introduced recently, generalizing the well-known univariate case. Numerous examples of such association schemes have already ...been exhibited. In particular, it has been demonstrated that the non-binary Johnson scheme is a bivariate P-polynomial association scheme. We show here that it is also a bivariate Q-polynomial association scheme for some parameters. This provides, with the P-polynomial structure, the bispectral property (i.e. the recurrence and difference relations) of a family of bivariate orthogonal polynomials made out of univariate Krawtchouk and dual Hahn polynomials. The algebra based on the bispectral operators is also studied together with the subconstituent algebra of this association scheme.
Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and for any integer i, let Γi(x) denote the set of vertices at distance i from x. ...Let V=ℂX denote the vector space over ℂ consisting of column vectors whose coordinates are indexed by X and whose entries are in ℂ, and for z∈X let ẑ denote the element of V with a 1 in the z coordinate and 0 in all other coordinates. Fix vertices x,u,v where u∈Γ2(x) and v∈Γ2(x)∩Γ2(u), and let T=T(x) denote the Terwilliger algebra with respect to x. Under certain additional combinatorial assumptions, we give a combinatorially-defined spanning set for a T-module of endpoint 2,and we give the action of the adjacency matrix on this spanning set. The vectors in our spanning set are defined as sums and differences of vectors ẑ, where the vertices z are chosen based on the their distances from x,u, and v.
We use this T-module to construct combinatorially-defined bases for all isomorphism classes of irreducible T-modules of endpoint 2 for examples including the Doubled Odd graphs, the Double Hoffman–Singleton graph, Tutte’s 12-cage graph, and the Foster graph. We provide a list of several other graphs satisfying our conditions.
Let Γ denote a finite, undirected, connected graph, with vertex set X. Fix a vertex x∈X. Associated with x is a certain subalgebra T=T(x) of MatX(C), called the subconstituent algebra. The algebra T ...is semisimple. Hora and Obata introduced a certain subalgebra Q⊆T, called the quantum adjacency algebra. The algebra Q is semisimple. In this paper we investigate how Q and T are related. In many cases Q=T, but this is not true in general. To clarify this issue, we introduce the notion of quasi-isomorphic irreducible T-modules. We show that the following are equivalent: (i) Q≠T; (ii) there exists a pair of quasi-isomorphic irreducible T-modules that have different endpoints. To illustrate this result we consider two examples. The first example concerns the Hamming graphs. The second example concerns the bipartite dual polar graphs. We show that for the first example Q=T, and for the second example Q≠T.
Let Γ denote a bipartite distance-regular graph with diameter at least 4 and valency at least 3. Fix a vertex of Γ and let T denote the corresponding Terwilliger algebra. Suppose that Γ is ...Q-polynomial and there are two non-isomorphic irreducible T-modules with endpoint 2. We show that, unless the intersection numbers of Γ fit one exceptional case (which is not known to correspond to an actual graph), the entry-wise product of pseudo primitive idempotents associated with these modules is a linear combination of two pseudo primitive idempotents. This result relates to a conjecture of MacLean and Terwilliger.
Let
Γ
denote a bipartite distance-regular graph with vertex set
X
, diameter
D
≥
4
, and valency
k
≥
3
. Let
C
X
denote the vector space over
C
consisting of column vectors with entries in
C
and rows ...indexed by
X
. For
z
∈
X
, let
z
^
denote the vector in
C
X
with a 1 in the
z
-coordinate, and 0 in all other coordinates. Fix a vertex
x
of
Γ
and let
T
=
T
(
x
)
denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible
T
-modules with endpoint 2, and they both are thin. Fix
y
∈
X
such that
∂
(
x
,
y
)
=
2
, where
∂
denotes path-length distance. For
0
≤
i
,
j
≤
D
define
w
i
j
=
∑
z
^
, where the sum is over all
z
∈
X
such that
∂
(
x
,
z
)
=
i
and
∂
(
y
,
z
)
=
j
. We define
W
=
span
{
w
i
j
∣
0
≤
i
,
j
≤
D
}
. In this paper we consider the space
M
W
=
span
{
m
w
∣
m
∈
M
,
w
∈
W
}
, where
M
is the Bose–Mesner algebra of
Γ
. We observe that
MW
is the minimal
A
-invariant subspace of
C
X
which contains
W
, where
A
is the adjacency matrix of
Γ
. We show that
4
D
-
6
≤
dim
(
M
W
)
≤
4
D
-
2
. We display a basis for
MW
for each of these five cases, and we give the action of
A
on these bases.
Let C denote the field of complex numbers, and fix a nonzero q∈C such that q4≠1. Define a C-algebra Δq by generators and relations in the following way. The generators are A, B, C. The relations ...assert that each ofA+qBC−q−1CBq2−q−2,B+qCA−q−1ACq2−q−2,C+qAB−q−1BAq2−q−2 is central in Δq. The algebra Δq is called the universal Askey–Wilson algebra. Let Γ denote a distance-regular graph that has q-Racah type. Fix a vertex x of Γ and let T=T(x) denote the corresponding subconstituent algebra. In this paper we discuss a relationship between Δq and T. Assuming that every irreducible T-module is thin, we display a surjective C-algebra homomorphism Δq→T. This gives a Δq action on the standard module of T.
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