Symmetric Graphs and Flag Graphs

 With any G-symmetric graph Γ admitting a nontrivial G-invariant partition , we may associate a natural “cross-sectional” geometry, namely the 1-design in which for and if and only if α is adjacent to at least one vertex in C, where and is the neighbourhood of B in the quotient graph of Γ with respect to . In a vast number of cases, the dual 1-design of contains no repeated blocks, that is, distinct vertices of B are incident in with distinct subsets of blocks of . The purpose of this paper is to give a general construction of such graphs, and then prove that it produces all of them. In particular, we show that such graphs can be reconstructed from and the induced action of G on . The construction reveals a close connection between such graphs and certain G-point-transitive and G-block-transitive 1-designs. By using this construction we give a characterization of G-symmetric graphs such that there is at most one edge between any two blocks of . This leads to, in a subsequent paper, a construction of G-symmetric graphs such that and each is incident in with vertices of B. The work was supported by a discovery-project grant from the Australian Research Council. Received April 24, 2001; in revised form October 9, 2002 Published online May 9, 2003     

Symmetric designs and geometroids

A-setS in a symmetric 2-(v, k, ) design is a subset which every block meets in 0, 1 or points such that for any point ofS there is a unique block meetingS at that point only. Ovoids in three-dimensional projective spaces are examples of-secs. It is shown that if has a-set then is a geometroid withv=u ²+u+1 andk=u+1, whereu–1. The cases whenu is–1, and+1 are investigated and some open problems discussed.     

Jordan Homomorphisms and Harmonic Mappings

 We show that each Jordan homomorphism R → R′ of rings gives rise to a harmonic mapping of one connected component of the projective line over R into the projective line over R′. If there is more than one connected component then this mapping can be extended in various ways to a harmonic mapping which is defined on the entire projective line over R. Received December 7, 2001; in revised form April 28, 2002 Published online January 7, 2003     

Tubes of even order and flat π.C 2 geometries

Atube of even orderq=2 ^d is a setT={L, } ofq+3 pairwise skew lines in PG(3,q) such that every plane onL meets the lines of in a hyperoval. Thequadric tube is obtained as follows. Take a hyperbolic quadricQ=Q ₃ ⁺ (q) in PG(3,q); letL be an exterior line, and let consist of the polar line ofL together with a regulus onQ.In this paper we show the existence of tubes of even order other than the quadric one, and we prove that the subgroup of PL(4,q) fixing a tube {L, } cannot act transitively on . As pointed out by a construction due to Pasini, this implies new results for the existence of flat .C ₂ geometries whoseC ₂-residues are nonclassical generalized quadrangles different from nets. We also give the results of some computations on the existence and uniqueness of tubes in PG(3,q) for smallq. Further, we define tubes for oddq (replacing hyperoval by conic in the definition), and consider briefly a related extremal problem.Dedicated to luigi antonio rosati on the occasion of his 70th birthday     

Translation Generalized Quadrangles In Even Characteristic

In this paper, we first introduce new objects called “translation generalized ovals” and “translation generalized ovoids”, and make a thorough study of these objects. We then obtain numerous new characterizations of the of Tits and the classical generalized quadrangle in even characteristic, including the complete classification of 2-transitive generalized ovals for the even case. Next, we prove a new strong characterization theorem for the of Tits. As a corollary, we obtain a purely geometric proof of a theorem of Johnson on semifield flocks. * The second author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium).     

Blocking Subspaces By Lines In PG(<Emphasis Type="Italic">n,q</Emphasis>)

This paper studies the cardinality of a smallest set of t-subspaces of the finite projective spaces PG(n, q) such that every s-subspace is incident with at least one element of , where 0 t < s n. This is a very difficult problem and the solution is known only for very few families of triples (s, t, n). When the answer is known, the corresponding blocking configurations usually are partitions of a subspace of PG(n, q) by subspaces of dimension t. One of the exceptions is the solution in the case t = 1 and n = 2s. In this paper, we solve the case when t = 1 and 2s < n 3s-3 and q is sufficiently large.     

Blockings-dimensional subspaces by lines inPG(2s,q)

We investigate sets of lines inPG(2s,q) such that everys-dimensional subspace contains a line of this set. We determine the minimum number of lines in such a set and show that there is only one type of such a set with this minimum number of lines.     

Teilmengengraphen und projektive Räume

Ohne Zusammenfassung

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Herrn Professor Dr. Janos Aczél zum 60. Geburtstag gewidmet     

Characterization of complete exterior sets of conics

Let be a set of exterior points of a nondegenerate conic inPG(2,q) with the property that the line joining any 2 points in misses the conic. Ifq1 (mod 4) then consists of the exterior points on a passant, ifq3 (mod 4) then other examples exist (at least forq=7, 11, ..., 31).Support from the Dutch organization for scientific Research (NWO) is gratefully acknowledged     

Self-dual sets of type (m, n) in projective spaces

In this paper we introduce and analyze the notion of self-dual k-sets of type (m, n). We show that in a non-square order projective space such sets exist only if the dimension is odd. We prove that, in a projective space of odd dimension and order q, self-dual k-sets of type (m, n), with , are of elliptic and hyperbolic type, respectively. As a corollary we obtain a new characterization of the non-singular elliptic and hyperbolic quadrics.     

Decomposable Near Polygons

We give a common construction for the product and the glued near polygons by generalizing the glueing construction given in [5]. We call the near polygons arising from this generalized glueing construction decomposable or (again) glued. We will study the geodetically closed sub near polygons of decomposable near polygons. Each decomposable near hexagon has a nice pair of partitions in geodetically closed near polygons. We will give a characterization of the decomposable near polygons using this property.     

A completion problem for finite affine planes

A partial affine plane (PAP) of ordern is ann ²-setS of points together with a collection ofn-subsets ofS called lines such that any two lines meet in at most one point. We obtain conditions under which a PAP with nearlyn ²+n lines can be completed to an affine plane by adding lines. In particular, we make use of Bruck’s completion condition for nets to show that certain PAP’s with at leastn ²+n−√n can be completed and that forn≠3 any PAP withn ²+n−2 lines can be completed.     

Minimal blocking sets of size 2p – 2 and 2p – 3 in PG(2, p), p prime and p > 5

A general construction of minimal blocking sets of size 2p – 3, where p is a prime and p ≡ 1 (mod 4), p > 5, and of size 2p – 2, where p is a prime and p ≡ 3 (mod 4), p > 5 in PG(2, p) is presented. These blocking sets are all of Rédei type.      

Möbius-Kantor configurations in the affine plane of order 7

We partition the affine plane of order 7 into a set of M?bius-Kantor configurations 8₃ plus a set consisting only of one point.     

Blocking Sets Of External Lines To A Conic In PG(2,q), q ODD

We determine all point-sets of minimum size in PG(2,q), q odd that meet every external line to a conic in PG(2,q). The proof uses a result on the linear system of polynomials vanishing at every internal point to the conic and a corollary to the classification theorem of all subgroups of PGL(2,q). * Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni and by the Hungarian-Italian Intergovernemental project “Algebraic and Geometric Structures”.