Type-II matrices in weighted Bose-Mesner algebras of ranks 2 and 3
A.D. Sankey
University of New Brunswick, Fredericton, NB E3B 5A3, Canada
DOI: 10.1007/s10801-009-0209-9
Abstract
Type-II matrices are nonzero complex matrices that were introduced in connection with spin models for link invariants. Type-II matrices have been found in connection with symmetric designs, sets of equiangular lines, strongly regular graphs, and some distance regular graphs. We investigate weighted complete and strongly regular graphs, and show that type-II matrices arise in this setting as well.
Pages: 133–153
Keywords: type-II matrix; association scheme; spin model
Full Text: PDF
References
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2. Bannai, E., Ito, T.: Algebraic Combinatorics. I: Association Schemes. Benjamin/Cummings, Menlo Park (1984)
3. Bussemaker, F.C., Mathon, R.A., Seidel, J.J.: Tables of two-graphs. In: Combinatorics and Graph Theory (Calcutta, 1980). Lecture Notes in Math., vol. 885, pp. 70-112. Springer, Berlin (1981)
4. Cameron, P.J., van Lint, J.H.: Designs, Graphs, Codes and Their Links. London Math. Soc. Student Texts, vol.
22. Cambridge University Press, Cambridge (1991)
5. Chan, A., Godsil, C.: Type-II matrices and combinatorial structures. [mathCO]
6. Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall Mathematics Series. Chapman & Hall, New York (1993)
7. Higman, D.G.: Coherent algebras. Linear Algebra Appl. 93, 209-239 (1987)
8. Higman, D.G.: Weights and t -graphs. Bull. Soc. Math. Belg. Sér. A 42(3), 501-521 (1990). Algebra, groups and geometry
9. Hosoya, R., Suzuki, H.: Type II matrices and their Bose-Mesner algebras. J. Algebraic Comb. 17(1), 19-37 (2003)
10. Jaeger, F., Matsumoto, M., Nomura, K.: Bose-Mesner algebras related to type II matrices and spin models. J. Algebraic Comb. 8(1), 39-72 (1998)
11. Jones, V.F.R.: On knot invariants related to some statistical mechanical models. Pac. J. Math. 137(2), 311-334 (1989)
12. Klin, M., Rücker, C., Rücker, G., Tinhofer, G.: Algebraic combinatorics in mathematical chemistry methods and algorithms. I. Permutation groups and coherent (cellular) algebras. MATCH Commun. Math. Comput. Chem. 40, 7-138 (1999)
13. Nomura, K.: An algebra associated with a spin model. J. Algebraic Comb. 6(1), 53-58 (1997)
14. Nomura, K.: Spin models and Bose-Mesner algebras. Eur. J. Comb. 20(7), 691-700 (1999)
15. Sankey, A.D.: Regular weights of full rank on strongly regular graphs. Isr. J. Math. 95, 1-23 (1996)
16. Sankey, A.D.: Regular weights and the Johnson scheme. Isr. J. Math. 97, 11-28 (1997)
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