Kazumasa Nomura

Distance degree regular graphs

Abstract In this note, inequalities between the distance degrees of distance degree regular graphs and to characterize the graphs when one of the equalities holds are proved.

Homogeneous graphs and regular near polygons

Abstract We consider a distance-regular graph having homogeneous edge patterns in each entry of its intersection diagram with respect to an edge. We call such graphs homogeneous graphs. We study elementary properties of homogeneous graphs, and we show these graphs are related deeply with regular near polygons.

Bose-Mesner Algebras Related to Type II Matrices and Spin Models

A type II matrix is a square matrixW with non-zero complex entries such that the entrywise quotient of any two distinct rows of W sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the construction, for every type II matrix W, of a Bose-Mesner algebraN(W) , which is a commutative algebra of matrices containing the identity I, the all-one matrix J, closed under transposition and under Hadamard (i.e., entrywise) product. Moreover, ifW is a spin model, it belongs to N(W). The transposition of matrices W corresponds to a classical notion of duality for the corresponding Bose-Mesner algebrasN(W) . Every Bose-Mesner algebra encodes a highly regular combinatorial structure called an association scheme, and we give an explicit construction of this structure. This allows us to compute N(W) for a number of examples.

Spin models on bipartite distance-regular graphs

Spin models were introduced by V. Jones (Pac. J. Math.137 (1989), 311-336) to construct invariants of knots and links. A spin model will be defined as a pair S = (X, w) of a finite set X and a function w on X × X satisfying several axioms. Some important spin models can be constructed on a distance-regular graph Γ = (X, E) with suitable complex numbers t0, t1, ..., td (d is the diameter of Γ) by putting w(a, b) = t∂(a, b). In this paper we determine bipartite distance-regular graphs which give spin models in this way with distinct t1, ..., td We show that such a bipartite distance-regular graph satisfies a strong regularity condition (it is 2-homogeneous), and we classify bipartite distance-regular graphs which satisfy this regularity condition.

A remark on the intersection arrays of distance-regular graphs

Abstract It is shown that the number of columns of type (1, 1, k − 2) in the intersection array of a distance-regular graph with valency k and girth >3 is at most four.

Distance-Regular Graphs Related to the Quantum Enveloping Algebra of sl(2)

AbstractWe investigate a connection between distance-regular graphs and Uq(sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let Γ be a distance-regular graph with diameter d ≥ 3 and valency k ≥ 3, and assume Γ is not isomorphic to the d-cube. Fix a vertex x of Γ, and let

Spin models constructed from Hadamard matrices

\mathcal{T} = \mathcal{T}(x)

Some Formulas for Spin Models on Distance-Regular Graphs

(x) denote the Terwilliger algebra of Γ with respect to x. Fix any complex number q ∉ {0, 1, −1}. Then