Q-polynomial Distance-regular Graph Research Articles

Correction to: Thin Q-Polynomial Distance-Regular Graphs Have Bounded $$c_{2}$$

Correction to: Thin Q-Polynomial Distance-Regular Graphs Have Bounded $$c_{2}$$

Remarks on pseudo-vertex-transitive graphs with small diameter

Let Γ denote a Q-polynomial distance-regular graph with vertex set X and diameter D. Let A denote the adjacency matrix of Γ. For a vertex x∈X and for 0≤i≤D, let Ei⁎(x) denote the projection matrix to the ith subconstituent space of Γ with respect to x. The Terwilliger algebra T(x) of Γ with respect to x is the semisimple subalgebra of MatX(C) generated by A,E0⁎(x),E1⁎(x),…,ED⁎(x). Let V denote a C-vector space consisting of complex column vectors with rows indexed by X. We say Γ is pseudo-vertex-transitive whenever for any vertices x,y∈X, there exists a C-vector space isomorphism ρ:V→V such that (ρA−Aρ)V=0 and (ρEi⁎(x)−Ei⁎(y)ρ)V=0 for all 0≤i≤D. In this paper, we discuss pseudo-vertex transitivity for distance-regular graphs with diameter D∈{2,3,4}. For D=2, we show that a strongly regular graph is pseudo-vertex-transitive if and only if all its local graphs have the same spectrum. For D=3, we consider the Taylor graphs and show that they are pseudo-vertex transitive. For D=4, we consider the antipodal tight graphs and show that they are pseudo-vertex transitive.

Designs in Finite Metric Spaces: A Probabilistic Approach

A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in Q-polynomial distance-regular graphs. An approximation of their cumulative distribution function, based on the notion of Christoffel function in approximation theory is given. As an application we derive limit laws on the weight distributions of binary orthogonal arrays of strength going to infinity. An analogous result for combinatorial designs of strength going to infinity is given.

The Norton algebra of a Q-polynomial distance-regular graph

We consider the Norton algebra associated with a Q-polynomial primitive idempotent of the adjacency matrix for a distance-regular graph. We obtain a formula for the Norton algebra product that we find attractive.

Connectivity concerning the last two subconstituents of a Q-polynomial distance-regular graph

Let Γ be a Q-polynomial distance-regular graph of diameter d≥3. Fix a vertex γ of Γ and consider the subgraph induced on the union of the last two subconstituents of Γ with respect to γ. We prove that this subgraph is connected.

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

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

A Group Commutator Involving the Last Distance Matrix and Dual Distance Matrix of a Q-Polynomial Distance-Regular Graph: The Hamming Graph Case

Let $$\varGamma $$ denote the Hamming graph H(D, r) with $$r \ge 3$$ . Consider the distance matrices $$\{A_i\}_{i=0}^{D}$$ of $$\varGamma $$ . Fix a vertex x of $$\varGamma $$ , and consider the dual distance matrices $$\{A_i^{*}\}_{i=0}^{D}$$ of $$\varGamma $$ with respect to x. We investigate the group commutator $$A_{D}^{-1}A_{D}^{*-1}A_{D}A_{D}^{*}$$ . We show that this matrix is diagonalizable. We compute its eigenvalues and their eigenspaces. Let T denote the subconstituent algebra of $$\varGamma $$ with respect to x. We describe the action of $$A_{D}^{-1}A_{D}^{*-1}A_{D}A_{D}^{*}$$ on each irreducible T-module.

Nonsymmetric Askey–Wilson polynomials and Q-polynomial distance-regular graphs

In his famous theorem (1982), Douglas Leonard characterized the q-Racah polynomials and their relatives in the Askey scheme from the duality property of Q-polynomial distance-regular graphs. In this paper we consider a nonsymmetric (or Laurent) version of the q-Racah polynomials in the above situation. Let Γ denote a Q-polynomial distance-regular graph that contains a Delsarte clique C. Assume that Γ has q-Racah type. Fix a vertex x∈C. We partition the vertex set of Γ according to the path-length distance to both x and C. The linear span of the characteristic vectors corresponding to the cells in this partition has an irreducible module structure for the universal double affine Hecke algebra Hˆq of type (C1∨,C1). From this module, we naturally obtain a finite sequence of orthogonal Laurent polynomials. We prove the orthogonality relations for these polynomials, using the Hˆq-module and the theory of Leonard systems. Changing Hˆq by Hˆq−1 we show how our Laurent polynomials are related to the nonsymmetric Askey–Wilson polynomials, and therefore how our Laurent polynomials can be viewed as nonsymmetric q-Racah polynomials.

Hypercube and tetrahedron algebra

Let D be an integer at least 3 and let H(D, 2) denote the hypercube. It is known that H(D, 2) is a Q-polynomial distance-regular graph with diameter D, and its eigenvalue sequence and its dual eigenvalue sequence are all {D − 2i}i=0D, Suppose that ⊠ denotes the tetrahedron algebra. In this paper, the authors display an action of ⊠ on the standard module V of H(D, 2). To describe this action, the authors define six matrices in MatX(ℂ), called $$A,A^* ,B,B^* ,K,K^* .$$ Moreover, for each matrix above, the authors compute the transpose and then compute the transpose of each generator of ⊠ on V.

Twice Q-polynomial distance-regular graphs of diameter 4

It is known that a distance-regular graph with valency $k$ at least three admits at most two Q-polynomial structures. % In this note we show that all distance-regular graphs with diameter four and valency at least three admitting two $Q$-polynomial structures are either dual bipartite or almost dual imprimitive. By the work of Dickie \cite{Dickie} this implies that any distance-regular graph with diameter $d$ at least four and valency at least three admitting two $Q$-polynomial structures is, provided it is not a Hadamard graph, either the cube $H(d,2)$ with $d$ even, the half cube ${1}/{2} H(2d+1,2)$, the folded cube $\tilde{H}(2d+1,2)$, or the dual polar graph on $[^2A_{2d-1}(q)]$ with $q\ge 2$ a prime power.

The Terwilliger polynomial of a Q-polynomial distance-regular graph and its application to pseudo-partition graphs

Let Γ be a Q-polynomial distance-regular graph with diameter at least 3. Terwilliger (1993) implicitly showed that there exists a polynomial, say T(λ)∈R[λ], of degree 4 depending only on the intersection numbers of Γ and its Q-polynomial ordering and such that T(η)≥0 holds for any non-principal eigenvalue η of the local graph Γ(x) for any vertex x∈V(Γ).We call T(λ) the Terwilliger polynomial of Γ. In this paper, we give an explicit formula for T(λ) in terms of the intersection numbers of Γ and the dual eigenvalues of Γ with respect to the first primitive idempotent in its Q-polynomial ordering. We then apply this polynomial to show that all pseudo-partition graphs with diameter at least 3 are known.

A characterization of bipartite Leonard pairs using the notion of a tail

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A⁎:V→V that satisfy (i) and (ii) below.(i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A⁎ is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A⁎ is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Very roughly speaking, a Leonard pair is a linear algebraic abstraction of a Q-polynomial distance-regular graph. There is a well-known class of distance-regular graphs said to be bipartite and there is a related notion of a bipartite Leonard pair. Recently, M.S. Lang introduced the notion of a tail for bipartite distance-regular graphs and there is an abstract version of this tail notion. Lang characterized the bipartite Q-polynomial distance-regular graphs using tails. In this paper, we obtain a similar characterization of the bipartite Leonard pairs using tails. Whereas Lang's arguments relied on the combinatorics of a distance-regular graph, our results are purely algebraic in nature.

Q-polynomial distance-regular graphs and a double affine Hecke algebra of rank one

We study a relationship between Q-polynomial distance-regular graphs and the double affine Hecke algebra of type (C1∨,C1). Let Γ denote a Q-polynomial distance-regular graph with vertex set X. We assume that Γ has q-Racah type and contains a Delsarte clique C. Fix a vertex x∈C. We partition X according to the path-length distance to both x and C. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a C-vector space W.The universal double affine Hecke algebra of type (C1∨,C1) is the C-algebra Hˆq defined by generators {tn±1}n=03 and relations (i) tntn−1=tn−1tn=1; (ii) tn+tn−1 is central; (iii) t0t1t2t3=q−1/2. In this paper, we display an Hˆq-module structure for W. For this module and up to affine transformation,•t0t1+(t0t1)−1 acts as the adjacency matrix of Γ;•t3t0+(t3t0)−1 acts as the dual adjacency matrix of Γ with respect to C;•t1t2+(t1t2)−1 acts as the dual adjacency matrix of Γ with respect to x. To obtain our results we use the theory of Leonard systems.

The Erdös-Ko-Rado theorem for twisted Grassmann graphs

We present a "modern" approach to the Erd\H{o}s-Ko-Rado theorem for Q-polynomial distance-regular graphs and apply it to the twisted Grassmann graphs discovered in 2005 by van Dam and Koolen.

A characterization of Leonard pairs using the parameters [formula omitted

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A*:V→V that satisfy (i) and (ii) below:(i)There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V. Pascasio recently obtained a characterization of the Q-polynomial distance-regular graphs using the intersection numbers ai. In this paper, we extend her results to a linear algebraic level and obtain a characterization of Leonard pairs. Pascasio’s argument appears to rely on the underlying combinatorial assumptions, so we take a different approach that is algebraic in nature.

Bipartite Q-polynomial distance-regular graphs and uniform posets

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 RL 2, LRL, L 2 R, 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 uniform structure. We show that except for one special case a uniform structure is attained, and except for three special cases a strongly uniform structure is attained.

Triple intersection numbers of [formula omitted]-polynomial distance-regular graphs

We use the system of linear Diophantine equations introduced by Coolsaet and Jurišić to obtain information about a feasible family of distance-regular graphs with vanishing Krein parameter q221 and intersection arrays {(r+1)(r3−1),r(r−1)(r2+r−1),r2−1;1,r(r+1),(r2−1)(r2+r−1)},r≥2. In this way we are able to calculate certain triple intersection numbers and prove nonexistence for all r≥3. For r=3 nonexistence was not known before, however it is well known that the intersection array for r=2 uniquely determines the halved 7-cube. Then we show how to apply Terwilliger balanced set conditions for Q-polynomial distance-regular graphs to produce additional linear Diophantine equations.

The Q-polynomial idempotents of a distance-regular graph

We obtain the following characterization of Q-polynomial distance-regular graphs. Let Γ denote a distance-regular graph with diameter d ⩾ 3 . Let E denote a minimal idempotent of Γ which is not the trivial idempotent E 0 . Let { θ i ∗ } i = 0 d denote the dual eigenvalue sequence for E. We show that E is Q-polynomial if and only if (i) the entry-wise product E ∘ E is a linear combination of E 0 , E, and at most one other minimal idempotent of Γ; (ii) there exists a complex scalar β such that θ i − 1 ∗ − β θ i ∗ + θ i + 1 ∗ is independent of i for 1 ⩽ i ⩽ d − 1 ; (iii) θ i ∗ ≠ θ 0 ∗ for 1 ⩽ i ⩽ d .

Structure of thin irreducible modules of a Q-polynomial distance-regular graph

Let Γ be a Q-polynomial distance-regular graph with vertex set X, diameter D ⩾ 3 and adjacency matrix A. Fix x ∈ X and let A ∗ = A ∗ ( x ) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T = T ( x ) is the subalgebra of Mat X ( C ) generated by A and A ∗ . Let W denote a thin irreducible T-module. It is known that the action of A and A ∗ on W induces a linear algebraic object known as a Leonard pair. Over the past decade, many results have been obtained concerning Leonard pairs. In this paper, we apply these results to obtain a detailed description of W. In our description, we do not assume that the reader is familiar with Leonard pairs. Everything will be proved from the point of view of Γ . Our results are summarized as follows. Let { E i } i = 0 D be a Q-polynomial ordering of the primitive idempotents of Γ and let { E i ∗ } i = 0 D be the dual primitive idempotents of Γ with respect to x. Let r , t and d be the endpoint, dual endpoint and diameter of W, respectively. Let u and v be nonzero vectors in E t W and E r ∗ W respectively. We show that { E r + i ∗ A i v } i = 0 d and { E t + i A * i u } i = 0 d are bases for W that are orthogonal with respect to the standard Hermitian dot product. We display the matrix representations of A and A ∗ with respect to these bases. We associate with W two sequences of polynomials { p i } i = 0 d and { p i ∗ } i = 0 d . We show that for 0 ⩽ i ⩽ d , p i ( A ) v = E r + i ∗ A i v and p i ∗ ( A ∗ ) u = E t + i A * i u . Next, we show that { E r + i ∗ u } i = 0 d and { E t + i v } i = 0 d are orthogonal bases for W ; we call these the standard basis and dual standard basis for W , respectively. We display the matrix representations of A and A ∗ with respect to these bases. The entries in these matrices will play an important role in our theory. We call these the intersection numbers and dual intersection numbers of W . Using these numbers, we compute all inner products involving the standard and dual standard bases. We also use these numbers to define two normalizations u i , v i (resp. u i ∗ , v i ∗ ) for p i (resp. p i ∗ ). Using the orthogonality of the standard and dual standard bases, we show that for each of the sequences { p i } i = 0 d , { p i ∗ } i = 0 d , { u i } i = 0 d , { u i ∗ } i = 0 d , { v i } i = 0 d , { v i ∗ } i = 0 d the polynomials involved are orthogonal and we display the orthogonality relations. We also show that each of the sequences satisfy a three-term recurrence and a relation known as the Askey–Wilson duality. We then turn our attention to two more bases for W . We find the matrix representations of A and A ∗ with respect to these bases. From the entries of these matrices we obtain two sequences of scalars known as the first split sequence and second split sequence of W . We associate with W a sequence of scalars called the parameter array. This sequence consists of the eigenvalues of the restriction of A to W, the eigenvalues of the restriction of A ∗ to W , the first split sequence of W and the second split sequence of W. We express all the scalars and polynomials associated with W in terms of its parameter array. We show that the parameter array of W is determined by r , t , d and one more free parameter. From this we conclude that the isomorphism class of W is determined by these four parameters. Finally, we apply our results to the case in which Γ has q-Racah type or classical parameters.

A characterization of Q-polynomial distance-regular graphs

Let Γ denote a distance-regular graph with diameter D ⩾ 3 . Let θ denote a nontrivial eigenvalue of Γ and let θ 0 * , θ 1 * , … , θ D * denote the corresponding dual eigenvalue sequence. In this paper we prove that Γ is Q-polynomial with respect to θ if and only if the following (i)–(iii) hold: (i) There exist β , γ * ∈ C such that (1) γ * = θ i - 1 * - β θ i * + θ i + 1 * ( 1 ⩽ i ⩽ D - 1 ) . (ii) There exist γ , ω , η * ∈ C such that the intersection numbers a i satisfy a i ( θ i * - θ i - 1 * ) ( θ i * - θ i + 1 * ) = γ θ i * 2 + ω θ i * + η * for 0 ⩽ i ⩽ D , where θ - 1 * and θ D + 1 * are the scalars which satisfy Eq. (1) for i = 0 , i = D , respectively. (iii) θ i * ≠ θ 0 * for 1 ⩽ i ⩽ D .