A subset of the vertex set of a graph G, S ⊆ V (G), is a (k, τ)-regular set if it induces a k-regular subgraph of G and every vertex not in the subset has τ neighbors in it. This paper is a contribution to the given problem of existence of (k, τ)-regular sets associated with all distinct eigenvalues of integral strongly regular graphs. The minimal idempotents of the Bose-Mesner algebra of stron...

We build interacting Fock spaces from association schemes and set up quantum walks on the resulting regular graphs (distance-regular distance-transitive). The construction is valid for growing space well defined asymptotically graph. To realize in terms of anyons we switch to dual view identify corresponding modular tensor categories Bose–Mesner algebra. Informally, fusion ring induced by schem...

Abstract. In 1989, Vaughan Jones introduced spin models and showed that they could be used to form link invariants in two different ways—by constructing representations of the braid group, or by constructing partition functions. These spin models were subsequently generalized to so-called 4-weight spin models by Bannai and Bannai; these could be used to construct partition functions, but did no...

By choosing a dynamical system with d different couplings, one can rearrange a system based on the graph with given vertex dependent on the dynamical system elements. The relation between the dynamical elements (coupling) is replaced by a relation between the vertexes. Based on the E0 transverse projection operator. We addressed synchronization problem of an array of the linearly coupled map la...

Let Γ denote a distance-regular graph with diameter D ≥ 3, intersection numbers ai, bi, ci and Bose-Mesner algebra M. For θ ∈ C ∪∞ we define a 1 dimensional subspace of M which we call M(θ). If θ ∈ C then M(θ) consists of those Y in M such that (A−θI)Y ∈ CAD, where A (resp. AD) is the adjacency matrix (resp. Dth distance matrix) of Γ. If θ = ∞ then M(θ) = CAD. By a pseudo primitive idempotent f...

Generalized table algebras were introduced in Arad, Fisman and Muzychuk (Israel J. Math. 114 (1999), 29–60) as an axiomatic closure of some algebraic properties of the Bose-Mesner algebras of association schemes. In this note we show that if all non-trivial degrees of a generalized integral table algebra are even, then the number of real basic elements of the algebra is bounded from below (Theo...

We study the four-weight spin models (W1, W2, W3, W4) introduced by Eiichi and Etsuko Bannai (Pacific J. of Math, to appear). We start with the observation, based on the concept of special link diagram, that two such spin models yield the same link invariant whenever they have the same pair (W1, W3), or the same pair (W2, W4). As a consequence, we show that the link invariant associated with a ...

Let Γ denote a bipartite Q-polynomial distance-regular graph with vertex set X, diameter d ≥ 3 and valency k ≥ 3. Let RX denote the vector space over R consisting of column vectors with entries in R and rows indexed by X. For z ∈ X, let ẑ denote the vector in RX with a 1 in the z-coordinate, and 0 in all other coordinates. Fix x, y ∈ X such that ∂(x, y) = 2, where ∂ denotes path-length distance...