On the isomorphism of certain primitive Q-polynomial not P-polynomial association schemes (original) (raw)

Pseudocyclic association schemes arising from the actions of and

Journal of Combinatorial Theory, Series A, 2006

The action of PGL(2, 2 m) on the set of exterior lines to a nonsingular conic in PG(2, 2 m) affords an association scheme, which was shown to be pseudocyclic in [6]. It was further conjectured in [6] that the orbital scheme of PΓL(2, 2 m) on the set of exterior lines to a nonsingular conic in PG(2, 2 m) is also pseudocyclic if m is an odd prime. We confirm this conjecture in this paper. As a by-product, we obtain a class of Latin square type strongly regular graphs on nonprime-power number of points.

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

European Journal of Combinatorics, 2002

Ito, Tanabe and Terwilliger recently introduced the notion of a tridiagonal pair. We apply their results to distance-regular graphs and obtain the following theorem. THEOREM. Let denote a distance-regular graph with diameter D ≥ 3. Suppose is Q-polynomial with respect to the ordering E 0 , E 1 ,. .. , E D of the primitive idempotents. For 0 ≤ i ≤ D, let m i denote the multiplicity of E i. Then (i) m i−1 ≤ m i (1 ≤ i ≤ D/2), (ii) m i ≤ m D−i (0 ≤ i ≤ D/2). By proving the above theorem we resolve a conjecture of Dennis Stanton.

On symmetric association schemes and associated quotient-polynomial graphs

Algebraic combinatorics, 2022

Let Γ denote an undirected, connected, regular graph with vertex set X, adjacency matrix A, and d + 1 distinct eigenvalues. Let A = A(Γ) denote the subalgebra of Mat X (C) generated by A. We refer to A as the adjacency algebra of Γ. In this paper we investigate algebraic and combinatorial structure of Γ for which the adjacency algebra A is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) A has a standard basis {I, F 1 ,. .. , F d }; (ii) for every vertex there exists identical distance-faithful intersection diagram of Γ with d + 1 cells; (iii) the graph Γ is quotient-polynomial; and (iv) if we pick F ∈ {I, F 1 ,. .. , F d } then F has d + 1 distinct eigenvalues if and only if span{I, F 1 ,. .. , F d } = span{I, F,. .. , F d }. We describe the combinatorial structure of quotient-polynomial graphs with diameter 2 and 4 distinct eigenvalues. As a consequence of the technique from the paper we give an algorithm which computes the number of distinct eigenvalues of any Hermitian matrix using only elementary operations. When such a matrix is the adjacency matrix of a graph Γ, a simple variation of the algorithm allow us to decide wheter Γ is distance-regular or not. In this context, we also propose an algorithm to find which distance-i matrices are polynomial in A, giving also these polynomials.

Cyclotomic Association Schemes and Strongly Regular Graphs

2008

Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence matrix of a symmetric design and the all-ones matrix. Amorphous pseudocyclic association schemes are examples of such association schemes whose associated symmetric design is trivial. We present several non-amorphous examples, which are either cyclotomic association schemes, or their fusion schemes. Special properties of symmetric designs guarantees the existence of further fusions, and the two known non-amorphous association schemes of class 4 discovered by van Dam and by the authors, are recovered in this way. We also give another pseudocyclic non-amorphous association scheme of class 7 on GF(2^{21}), and a new pseudocyclic amorphous association scheme of class 5 on GF(2^{12}).

The Terwilliger algebra of an almost-bipartite P-and Q-polynomial association scheme

Discrete mathematics, 2005

Let Y denote a D-class symmetric association scheme with D ≥ 3, and suppose Y is almostbipartite P-and Q-polynomial. Let x denote a vertex of Y and let T = T (x) denote the corresponding Terwilliger algebra. We prove that any irreducible T -module W is both thin and dual thin in the sense of Terwilliger. We produce two bases for W and describe the action of T on these bases. We prove that the isomorphism class of W as a T -module is determined by two parameters, the dual endpoint and diameter of W . We find a recurrence which gives the multiplicities with which the irreducible T -modules occur in the standard module. We compute this multiplicity for those irreducible T -modules which have diameter at least D − 3.

Spectra of bipartite P-and Q-polynomial association schemes

Graphs and Combinatorics, 1998

Let Y = (X, {R i } 0≤i≤D ) denote a symmetric association scheme with D ≥ 3, and assume Y is not an ordinary cycle. Suppose Y is bipartite Ppolynomial with respect to the given ordering A 0 , A 1 , ..., A D of the associate matrices, and Q-polynomial with respect to the ordering E 0 , E 1 , ..., E D of the primitive idempotents. Then the eigenvalues and dual eigenvalues satisfy exactly one of (i) -(iv).

On Bipartite QQQ-Polynomial Distance-Regular Graphs with c_2le2c_2 \le 2c_2le2

Electronic Journal of Combinatorics, 2014

Let Γ denote a bipartite Q-polynomial distance-regular graph with diameter D 4, valency k 3 and intersection number c 2 2. We show that Γ is either the Ddimensional hypercube, or the antipodal quotient of the 2D-dimensional hypercube, or D = 5.