A Leonard triple on a vector space V is a triple of linear operators on V such that for each of these operators there is a basis of V with respect to which the matrix representing it is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let Q be the graph of the D-hypercube. Let A, A* and T be the adjacency matrix, the dual adjacency matrix of Q, and the algebra generated by A, A*. We show that there exists a matrix A^e such that (A, A*, A^e) acts on each irreducible T-module as a Leonard triple. We give a detailed description of these Leonard triples.
COBISS.SI-ID: 14624857
Let G denote a Q-polynomial distance-regular graph with diameter D > 2 and intersection numbers a_1=0, a_2 \ne 0. Let T denote the Terwilliger algebra of G. We show that up to isomorphism there exists a unique irreducible T-module W with endpoint 1. We show that W has dimension 2D-2. We display a basis for W which consists of eigenvectors for A*. We display the action of A on this basis. We show that W appears in the standard module of G with multiplicity k-1, where k is the valency of G.
COBISS.SI-ID: 14627929
Let G denote a distance-regular graph with diameter D >2. Assume G has classical parameters (D,b,\alpha,\beta) with b < -1. Let T denote the Terwilliger algebra of G. We show that up to isomorphism there exist exactly two irreducible T-modules with endpoint 1; their dimensions are D and 2D-2. For these T-modules we display a basis consisting of eigenvectors for A*, and for each basis we give the action of A.
COBISS.SI-ID: 2132965
Let G denote a bipartite Q-polynomial distance-regular graph. Fix vertices x,y which are at distance 2. Let w_{ij} be the characteristic vector of the set of vertices, which are at distance i from x and at distance j from y. Let W=span{w_{ij} | 0 \le i,j \le d}. We consider the space MW=span{mw | m in M, w in W}, where M is the Bose-Mesner algebra of G. We display an orthogonal basis for MW. We give the action of A on this basis. We show that the dimension of MW is 3d-3 if G is 2-homogeneous, 3d-1if G is the antipodal quotient of the 2d-cube, and 4d-4 otherwise.
COBISS.SI-ID: 1796823
The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles of even length. It is proved that every non-trivial distance-regular Cayley graph on a dihedral group is bipartite, non-antipodal, has diameter 3 and arises either from a cyclic difference set, or possibly (if any such exists) from a dihedral difference set satisfying some additional conditions.
COBISS.SI-ID: 1909207