An Algebra Associated with a Spin Model
Journal of Algebraic Combinatorics: An International Journal
On Spin Models, Modular Invariance, and Duality
Journal of Algebraic Combinatorics: An International Journal
Bose-Mesner Algebras Related to Type II Matrices and Spin Models
Journal of Algebraic Combinatorics: An International Journal
Duality Maps of Finite Abelian Groups and Their Applications to Spin Models
Journal of Algebraic Combinatorics: An International Journal
Symmetric Versus Non-Symmetric Spin Models for Link Invariants
Journal of Algebraic Combinatorics: An International Journal
On Four-Weight Spin Models and their Gauge Transformations
Journal of Algebraic Combinatorics: An International Journal
Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras
Journal of Algebraic Combinatorics: An International Journal
Spin models, association schemes and the Nakanishi-Montesinos conjecture
European Journal of Combinatorics
Homogeneity of a Distance-Regular Graph Which Supports a Spin Model
Journal of Algebraic Combinatorics: An International Journal
Algebraic Characterizations of Graph Regularity Conditions
Designs, Codes and Cryptography
Terwilliger algebras of wreath products of one-class association schemes
Journal of Algebraic Combinatorics: An International Journal
Coherent configurations and triply regular association schemes obtained from spherical designs
Journal of Combinatorial Theory Series A
Classification of commutative association schemes with almost commutative Terwilliger algebras
Journal of Algebraic Combinatorics: An International Journal
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Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using series-parallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.