By Bart De Bruyn

This booklet supplies an advent to the sector of occurrence Geometry via discussing the fundamental households of point-line geometries and introducing the various mathematical ideas which are crucial for his or her research. The households of geometries lined during this ebook comprise between others the generalized polygons, close to polygons, polar areas, twin polar areas and designs. additionally a number of the relationships among those geometries are investigated. Ovals and ovoids of projective areas are studied and a few functions to specific geometries might be given. A separate bankruptcy introduces the mandatory mathematical instruments and strategies from graph conception. This bankruptcy itself may be considered as a self-contained creation to strongly typical and distance-regular graphs.

This publication is basically self-contained, merely assuming the information of uncomplicated notions from (linear) algebra and projective and affine geometry. just about all theorems are observed with proofs and a listing of routines with complete suggestions is given on the finish of the ebook. This e-book is geared toward graduate scholars and researchers within the fields of combinatorics and occurrence geometry.

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Ordinary n-gons with n ≥ 5 odd are examples ). The notion of Moore of Moore graphs (with valency 2 and diameter n−1 2 geometry is due to Bose and Dowling [20]. The notion of a generalized Moore geometry is due to Roos and van Zanten [109]. Regarding Moore graphs, the following classification results are known. 5 ([5, 46]) A Moore graph of diameter d ≥ 3 is an ordinary (2d + 1)-gon. 6 ([84]) A Moore graph of diameter 2 has valency 2, 3, 7 or 57. Up to isomorphism, the pentagon is the unique Moore graph of diameter 2 and valency 2.

We will prove the following facts in Chapter 5: • a generalized n-gon, n ≥ 2, has diameter n 2 ; • every generalized n-gon, n ≥ 3, is a near n-gon; • the double of a generalized n-gon, n ≥ 2, is a generalized 2n-gon. In the following two theorems, we collect some restrictions that must be satisfied by orders of finite generalized polygons. These results will also be proved in Chapter 5. The first theorem is due to Feit and Higman [60]. 3 ([60]) Let S be a finite generalized n-gon, n ≥ 3, of order (s, t).

If m = 2n − 1, then there exists a reference system in PG(m, F ) with reσ σ spect to which H has equation (X0 X1σ +X1 X0σ )+· · ·+(Xm−1 Xm +Xm Xm−1 )= 0. In this case, we denote H (and (H, Σ)) also by H(2n − 1, F /F). If m = 2n, then there exists a reference system in PG(m, F ) with reσ + spect to which H has equation X0σ X0 + (X1 X2σ + X2 X1σ ) + · · · + (Xm−1 Xm σ Xm Xm−1 ) = 0. In this case, we denote H (and (H, Σ)) also by H(2n, F /F). In the finite case, we always have m ∈ {2n − 1, 2n}. If F ∼ = Fq and 2 , then we denote H(2n − 1, F /F) and H(2n, F /F) also by hence F ∼ F = q 2 2 H(2n − 1, q ) and H(2n, q ), respectively.

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