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On generalized Moore geometries. I, II. (English) Zbl 0626.51006
The author finds a bound for the diameter d of a nontrivial generalized Moore geometry.
Using formulas from E. Bannai and T. Ito [“Algebraic Combinatorics. I: Association schemes.” (1984; Zbl 0555.05019)], in the first part he proves that the minimal polynomial must factor into factors of degree at most 3 over integers. Then he applies the rationalization method of R. M. Damerell and M. A. Georgiacodis [J. Lond. Math. Soc., II. Ser. 23, 1-9 (1981; Zbl 0467.05019)] followed by reduction \(modulo 2\) and \(modulo 3.\) He obtains a list of exponential diophantine equations, each of which turns out to have a finite number of solutions. The largest of these solutions is \(d=161.\)
In the second part the diameter d of a generalized Moore geometry is shown to be at most 13. He applies various ad hoc methods on a case - by - case method to lower the bound. The author means that the bound 13 could be further reduced, although it seems difficult.
Reviewer: P.Burda

51C05 Ring geometry (Hjelmslev, Barbilian, etc.)
51E99 Finite geometry and special incidence structures
05B25 Combinatorial aspects of finite geometries
Full Text: DOI
[1] Bannai, E.; Ito, T., Algebraic combinatories, I, (1984), Benjamin New York
[2] Damerell, R.M.; Georgiakodis, M.A., On the maximum diameter of a class of distance-regular graphs, Bull. London math. soc., 13, 316-322, (1981) · Zbl 0457.05055
[3] Damerell, R.M.; Georgiakodis, M.A., On Moore geometries, I, J. London math. soc., 23, 2, 1-9, (1981) · Zbl 0467.05019
[4] Damerell, R.M., On Moore geometries, II, (), 33-40 · Zbl 0467.05020
[5] Feit, W.; Higman, G., The nonexistence of certain generalized polygons, J. algebra, 1, 114-131, (1964) · Zbl 0126.05303
[6] Fuglister, F.J., On finite Moore geometries, J. combin. theory ser. A, 23, 187-197, (1977) · Zbl 0422.05021
[7] Fuglister, F.J., The nonexistence of Moore geometries of diameter 4, Discrete math., 45, 229-238, (1983) · Zbl 0511.05018
[8] Roos, C.; van Zanten, A.J., On the existence of certain distance-regular graphs, J. combin. theory ser. B, 33, 197-212, (1982) · Zbl 0488.05055
[9] Roos, C.; van Zanten, A.J., On the existence of certain generalized Moore geometries, I, Discrete math., 51, 179-190, (1984) · Zbl 0547.05021
[10] Ross, C.; van Zanten, A.J., On the existence of certain generalized Moore geometries, II, Discrete math., 51, 277-282, (1984) · Zbl 0547.05022
[11] van Zanten, A.J., The degree of the eigenvalues of generalized Moore geometries, Discrete math., 67, 89-96, (1987) · Zbl 0671.05061
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