# zbMATH — the first resource for mathematics

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

##### MSC:
 51C05 Ring geometry (Hjelmslev, Barbilian, etc.) 51E99 Finite geometry and special incidence structures 05B25 Combinatorial aspects of finite geometries
##### Keywords:
diameter; generalized Moore geometry
Full Text:
##### References:
  Bannai, E.; Ito, T., Algebraic combinatories, I, (1984), Benjamin New York  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  Damerell, R.M.; Georgiakodis, M.A., On Moore geometries, I, J. London math. soc., 23, 2, 1-9, (1981) · Zbl 0467.05019  Damerell, R.M., On Moore geometries, II, (), 33-40 · Zbl 0467.05020  Feit, W.; Higman, G., The nonexistence of certain generalized polygons, J. algebra, 1, 114-131, (1964) · Zbl 0126.05303  Fuglister, F.J., On finite Moore geometries, J. combin. theory ser. A, 23, 187-197, (1977) · Zbl 0422.05021  Fuglister, F.J., The nonexistence of Moore geometries of diameter 4, Discrete math., 45, 229-238, (1983) · Zbl 0511.05018  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  Roos, C.; van Zanten, A.J., On the existence of certain generalized Moore geometries, I, Discrete math., 51, 179-190, (1984) · Zbl 0547.05021  Ross, C.; van Zanten, A.J., On the existence of certain generalized Moore geometries, II, Discrete math., 51, 277-282, (1984) · Zbl 0547.05022  van Zanten, A.J., The degree of the eigenvalues of generalized Moore geometries, Discrete math., 67, 89-96, (1987) · Zbl 0671.05061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.