Spin Models and Almost Bipartite 2-Homogeneous Graphs

Nomura, Kazumasa

doi:10.2969/aspm/02410285

Cited by 9 publications

(11 citation statements)

References 18 publications

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“…In this section, we show the concept of tight is closely related to the concept of 1-homogeneous that appears in the work of Nomura [14][15][16]. …”

Section: The 1-homogeneous Propertymentioning

confidence: 95%

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Jurišić

Koolen

Terwilliger

2000

Journal of Algebraic Combinatorics

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Abstract. We consider a distance-regular graph with diameter d ≥ 3 and eigenvaluesWe show the intersection numbers a 1 , b 1 satisfyWe say is tight whenever is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show is tight if and only if the intersection numbers are given by certain rational expressions involving d independent parameters. We show is tight if and only if a 1 = 0, a d = 0, and is 1-homogeneous in the sense of Nomura. We show is tight if and only if each local graph is connected strongly-regular, with nontrivial eigenvalues −1 − b 1 (1 + θ 1 ) −1 and −1 − b 1 (1 + θ d ) −1 . Three infinite families and nine sporadic examples of tight distance-regular graphs are given.

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“…In this section, we show the concept of tight is closely related to the concept of 1-homogeneous that appears in the work of Nomura [14][15][16]. …”

Section: The 1-homogeneous Propertymentioning

confidence: 95%

“…Our third characterization of the tight condition involves the concept of 1-homogeneous that appears in the work of Nomura [14][15][16]. See also Curtin [7].…”

Section: Introductionmentioning

confidence: 99%

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Jurišić

Koolen

Terwilliger

2000

Journal of Algebraic Combinatorics

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“…Moreover, since 1 is self-dual, it has a nontrivial eigenvalue with multiplicity precisely b 0 . But it was shown in [29] that all bipartite and all almost bipartite distance-regular graphs with a nontrivial eigenvalue of multiplicity at most b 0 are 2-homogeneous.…”

Section: And1mentioning

confidence: 98%

“…The 2-homogeneous bipartite and 2-homogeneous almost bipartite distance-regular graphs were characterized in [29]. There are three families with arbitrary diameter the cycles, the Hamming cubes, and the folded Hamming cubes.…”

Section: And1mentioning

confidence: 99%

“…In looking for examples of spin models, it has often proved convenient to look at special classes of association schemes, such as those on few points [7], abelian group schemes [1,2,5,8], and on distance-regular graphs. Known examples in the distance-regular graph case include certain strongly regular graphs [18] (see also [14]), Hadamard graphs [26] (see also [19,21]), 2-homogeneous (almost) bipartite distance-regular graphs [27,28,29], and Hamming graphs [4]. (Section 9 contains a list of all examples of spin models, known to the authors, which lie in the Bose Mesner algebra of a distance-regular graph.)…”

Section: Introductionmentioning

confidence: 99%

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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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Spin Models and Bose–Mesner Algebras

Nomura

1999

European Journal of Combinatorics

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A type II matrix is a square matrix W 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 algebra N (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, if W 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 algebras N (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.

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Spin Models and Almost Bipartite 2-Homogeneous Graphs

Cited by 9 publications

References 18 publications

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Some Formulas for Spin Models on Distance-Regular Graphs

Spin Models and Bose–Mesner Algebras

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