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Let Γ = (X, R) denote a distance-regular graph with distance function ∂ and diameter d ≥ 3. For 2 ≤ i ≤ d, by a parallelogram of length i, we mean a 4-tuple xyzu of vertices in X such that ∂(x, y) = Suppose the intersection number a 1 = 0, a 2 = 0 in Γ. We prove the following (i)-(ii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of(More)
The quantum algebra U q (sl 2) and its equitable presentation * Abstract We show that the quantum algebra U q (sl 2) has a presentation with generators x ±1 , y, z and relations xx −1 = x −1 x = 1, qxy − q −1 yx q − q −1 = 1, qyz − q −1 zy q − q −1 = 1, qzx − q −1 xz q − q −1 = 1. We call this the equitable presentation. We show that y (resp. z) is not(More)
Let G be a simple connected graph of order n with degree sequence d 1 , d 2 , · · · , d n in non-increasing order. The spectral radius ρ(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer at most n, we give a sharp upper bound for ρ(G) by a function of d 1 , d 2 , · · · , d , which generalizes a series of previous results.
A superimposed code with general distance D can be used to construct a non-adaptive pooling design. It can then be used to identify a few unknown positives from a large set of items by associating naturally an outcome vector u. A simple method for decoding the outcome vector u is given whenever there are at most D−1 2 errors occuring in the outcome vector(More)
Let G = (V G, EG) be a connected graph on n vertices, with diameter D, adjacency matrix A, and distance function ∂. Assume that A has d + 1 distinct eigenvalues λ 0 > λ 1 > · · · > λ d with corresponding multiplicities m 0 = 1, m 1 ,. . ., m d. From the spectrum of G we then define an inner product ·, ·· on the vector space R d [x] of real polynomials of(More)
Let W denote a simply-laced Coxeter group with n generators. We construct an n-dimensional representation φ of W over the finite field F 2 of two elements. The action of φ(W) on F n 2 by left multiplication is corresponding to a combinatorial structure extracted and generalized from Vogan diagrams. In each case W of types A, D and E, we determine the orbits(More)
Let Γ denote a near polygon distance-regular graph with diameter d ≥ 3, valency k and intersection numbers a 1 > 0, c 2 > 1. Let θ 1 denote the second largest eigenvalue of Γ. We show θ 1 ≤ k − a 1 − c 2 c 2 − 1. We show the following (i)–(iii) are equivalent. (i) Equality is attained above; (ii) Γ is Q-polynomial with respect to θ 1 ; (iii) Γ is a dual(More)
Let Γ denote a distance-regular graph with diameter D ≥ 3, intersection numbers a i , b i , c i 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 ∈ CA D , where A (resp. A D) is the adjacency matrix (resp. Dth distance matrix) of Γ. If θ = ∞ then(More)