Hypergroups and distance distributions of random walks on graphs
DOI:
https://doi.org/10.7146/math.scand.a-122932Abstract
Wildberger's construction enables us to obtain a hypergroup from a random walk on a special graph. We will give a probability theoretic interpretation to products on the hypergroup. The hypergroup can be identified with a commutative algebra whose basis is transition matrices. We will estimate the operator norm of such a transition matrix and clarify a relationship between their matrix products and random walks.
References
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Jewett, R. I., Spaces with an abstract convolution of measures, Advances in Math. 18 (1975), no. 1, 1–101. https://doi.org/10.1016/0001-8708(75)90002-X
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Lasser, R., Orthogonal polynomials and hypergroups, Rend. Mat. (7) 3 (1983), no. 2, 185–209.
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Matsuzawa, Y., Ohno, H., Suzuki, A., Tsurii, T., and Yamanaka, S., Non-commutative hypergroup of order five, J. Algebra Appl. 16 (2017), no. 7, 1750127, 21 pp. https://doi.org/10.1142/S0219498817501274
Spector, R., Mesures invariantes sur les hypergroupes, Trans. Amer. Math. Soc. 239 (1978), 147–165. https://doi.org/10.2307/1997851
Wildberger, N. J., Hypergroups associated to random walks on Platonic solids, preprint Univ. of NSW, 1994.
Wildberger, N. J., Finite commutative hypergroups and applications from group theory to conformal field theory, in “Applications of hypergroups and related measure algebras (Seattle, WA, 1993)”, Contemp. Math., vol. 183, Amer. Math. Soc., Providence, RI, 1995, pp. 413–434. https://doi.org/10.1090/conm/183/02075
Wildberger, N. J., Strong hypergroups of order three, J. Pure Appl. Algebra 174 (2002), no. 1, 95–115. https://doi.org/10.1016/S0022-4049(02)00016-6
Bose, R. C. and Mesner, D. M., On linear associative algebras corresponding to association schemes of partially balanced designs, Ann. Math. Statist. 30 (1959), 21–38. https://doi.org/10.1214/aoms/1177706356
Brémaud, P., Markov chains: Gibbs fields, Monte Carlo simulation, and queues, Texts in Applied Mathematics, vol. 31, Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4757-3124-8
Brouwer, A. E., Cohen, A. M., and Neumaier, A., Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 18, Springer-Verlag, Berlin, 1989. https://doi.org/10.1007/978-3-642-74341-2
van Dam, E. R., Koolen, H., and Tanaka, H., Distance-regular graphs, Electron. J. Combin. Dynamic surveys (2016), #DS22, 156 pp. https://doi.org/10.37236/4925
Dunkl, C. F., The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc. 179 (1973), 331–348. https://doi.org/10.2307/1996507
Godsil, C. and Royle, G., Algebraic graph theory, Graduate Texts in Mathematics, vol. 207, Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4613-0163-9
Godsil, C. D., More odd graph theory, Discrete Math. 32 (1980), no. 2, 205–207. https://doi.org/10.1016/0012-365X(80)90055-2
Ikkai, T. and Sawada, Y., Hypergroups derived from random walks on some infinite graphs, Monatsh. Math. 189 (2019), no. 2, 321–353. https://doi.org/10.1007/s00605-018-1255-y
Jewett, R. I., Spaces with an abstract convolution of measures, Advances in Math. 18 (1975), no. 1, 1–101. https://doi.org/10.1016/0001-8708(75)90002-X
Krebs, M. and Shaheen, A., Expander families and Cayley graphs, Oxford University Press, Oxford, 2011.
Lasser, R., Orthogonal polynomials and hypergroups, Rend. Mat. (7) 3 (1983), no. 2, 185–209.
Lasser, R., Discrete commutative hypergroups, in “Inzell Lectures on Orthogonal Polynomials”, Adv. Theory Spec. Funct. Orthogonal Polynomials, vol. 2, Nova Sci. Publ., Hauppauge, NY, 2005, pp. 55–102.
Matsuzawa, Y., Ohno, H., Suzuki, A., Tsurii, T., and Yamanaka, S., Non-commutative hypergroup of order five, J. Algebra Appl. 16 (2017), no. 7, 1750127, 21 pp. https://doi.org/10.1142/S0219498817501274
Spector, R., Mesures invariantes sur les hypergroupes, Trans. Amer. Math. Soc. 239 (1978), 147–165. https://doi.org/10.2307/1997851
Wildberger, N. J., Hypergroups associated to random walks on Platonic solids, preprint Univ. of NSW, 1994.
Wildberger, N. J., Finite commutative hypergroups and applications from group theory to conformal field theory, in “Applications of hypergroups and related measure algebras (Seattle, WA, 1993)”, Contemp. Math., vol. 183, Amer. Math. Soc., Providence, RI, 1995, pp. 413–434. https://doi.org/10.1090/conm/183/02075
Wildberger, N. J., Strong hypergroups of order three, J. Pure Appl. Algebra 174 (2002), no. 1, 95–115. https://doi.org/10.1016/S0022-4049(02)00016-6
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Published
2021-02-17
How to Cite
Endo, K., Mimura, I., & Sawada, Y. (2021). Hypergroups and distance distributions of random walks on graphs. MATHEMATICA SCANDINAVICA, 127(1), 43–62. https://doi.org/10.7146/math.scand.a-122932
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