Characterization of H(n, q) by the parameters

Egawa, Yoshimi

doi:10.1016/0097-3165(81)90007-8

Cited by 59 publications

(34 citation statements)

References 4 publications

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“…By Lemma 5.3, we need not treat the case p=&1, so suppose p=1. Then the intersection numbers given in Lemma 5.2 are exactly those of the Hamming graph H(d, q), so that the graph is isomorphic to H(d, q) or the Doob graph by [31,15]. Now suppose that a 1 =0.…”

Section: And1mentioning

confidence: 92%

Some Formulas for Spin Models on Distance-Regular Graphs

Curtin

Nomura

1999

Journal of Combinatorial Theory, Series B

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A spin model is a square matrix W satisfying certain conditions which ensure that it yields an invariant of knots and links via a statistical mechanical construction of V. F. R. Jones. Recently F. Jaeger gave a topological construction for each spin model W of an association scheme which contains W in its Bose Mesner algebra. Shortly thereafter, K. Nomura gave a simple algebraic construction of such a Bose Mesner algebra N(W). In this paper we study the case W # A N(W), where A is the Bose Mesner algebra of a distance-regular graph. We show the following results. Let 1=(X, R) be a distance-regular graph of diameter d>1 such that the Bose Mesner algebra A of 1 satisfies W # A N(W) for some spin model W on X. Write W=

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Section: And1mentioning

confidence: 92%

Some Formulas for Spin Models on Distance-Regular Graphs

Curtin

Nomura

1999

Journal of Combinatorial Theory, Series B

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“…If q = 1, then by [8,Lemma 5.2] has the parameters of a Hamming graph. So by [11] is isomorphic to a Hamming graph or an Egawa graph. But by [7,Theorem 3.8], is thin and so by [16], is not an Egawa graph.…”

Section: The Parameters Of γmentioning

confidence: 99%

The Terwilliger Algebra of a Distance-Regular Graph that Supports a Spin Model

Caughman

Wolff

2005

J Algebr Comb

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Let denote a distance-regular graph with vertex set X , diameter D ≥ 3, valency k ≥ 3, and assume supports a spin modelTo avoid degenerate situations we assume is not a Hamming graph and t i ∈ {t 0 , −t 0 } for 1 ≤ i ≤ D. In an earlier paper Curtin and Nomura determined the intersection numbers of in terms of D and two complex parameters η and q. We extend their results as follows. Fix any vertex x ∈ X and let T = T (x) denote the corresponding Terwilliger algebra. Let U denote an irreducible T -module with endpoint r and diameter d. We obtain the intersection numbers c i (U ), b i (U ), a i (U ) as rational expressions involving r, d, D, η and q. We show that the isomorphism class of U as a T -module is determined by r and d. We present a recurrence that gives the multiplicities with which the irreducible T -modules appear in the standard module. We compute these multiplicites explicitly for the irreducible T -modules with endpoint at most 3. We prove that the parameter q is real and we show that if is not bipartite, then q > 0 and η is real.

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“…There are many contributions to this problem (e.g. [5] for the Hamming schemes, [11,14] for the Johnson schemes, [13] for the q-analogue Johnson schemes, [9] for the dual polar schemes, [8,10] for the forms schemes and so on), but almost all of them concern P-(and Q-) polynomial schemes.…”

Section: Introductionmentioning

confidence: 99%