The smallest split Cayley hexagon has two symplectic embeddings

AbstractIt is well known that the smallest split Cayley generalized hexagon H(2) can be embedded into the symplectic space W(5,2), or equivalently, into the parabolic quadric Q(6,2). We establish a second way to embed H(2) into the same space and describe a computer proof of the fact that these are essentially the only two embeddings of this type

Spherical quadrangles with three equal sides and rational angles

When the condition of having three equal sides is imposed upon a (convex) spherical quadrangle, the four angles of that quadrangle cannot longer be freely chosen but must satisfy an identity. We derive two simple identities of this kind, one involving ratios of sines, and one involving ratios of tangents, and improve upon an earlier identity by Ueno and Agaoka. The simple form of these identities enable us to further investigate the case in which all of the angles are rational multiples of pi and produce a full classification, consisting of 7 infinite classes and 29 sporadic examples. Apart from being interesting in its own right, these quadrangles play an important role in the study of spherical tilings by congruent quadrangles

Using equality in the Krein conditions to prove nonexistence of certain distance-regular graphs

AbstractWe prove the nonexistence of a distance-regular graph with intersection array {74,54,15;1,9,60} and of distance-regular graphs with intersection arrays{4r3+8r2+6r+1,2r(r+1)(2r+1),2r2+2r+1;1,2r(r+1),(2r+1)(2r2+2r+1)} with r an integer and r⩾1. Both cases serve to illustrate a technique which can help in determining structural properties for distance-regular graphs and association schemes with a sufficient number of vanishing Krein parameters

Arcs with large conical subsets in Desarguesian planes of even order

We give an explicit classification of the arcs in PG (2, q) (q even) with a large conical suset and excess 2, i.e., that consist of q/2 + 1 points of a conic and two points not on that conic. Apart from the initial setup,the methods used are similar to those for the case of odd q, published earlier

Generation of local symmetry-preserving operations on polyhedra

We introduce a new practical and more general definition of local symmetry-preserving operations on polyhedra. These can be applied to arbitrary embedded graphs and result in embedded graphs with the same or higher symmetry. With some additional properties we can restrict the connectivity, e.g. when we only want to consider polyhedra. Using some base structures and a list of 10 extensions, we can generate all possible local symmetry-preserving operations isomorph-free

The known maximal partial ovoids of size q2−1q^2-1 of Q(4,q)

We present a description of maximal partial ovoids of size $q^2-1$ of the parabolic quadric \q(4,q) as sharply transitive subsets of \SL(2,q) and show their connection with spread sets. This representation leads to an elegant explicit description of all known examples. We also give an alternative representation of these examples which is related to root systems.Comment: 15 pages, revised version (v2 on arxiv is just a update of another paper, applied on this paper, clearly a mistake...

Local orientation-preserving symmetry preserving operations on polyhedra

Unifying approaches by amongst others Archimedes, Kepler, Goldberg, Caspar and Klug, Coxeter, and Conway, and extending on a previous formalisation of the concept of local symmetry preserving (lsp) operations, we introduce a formal definition of local operations on plane graphs that preserve orientation-preserving symmetries, but not necessarily orientation-reversing symmetries. This operations include, e.g., the chiral Goldberg and Conway operations as well as all lsp operations. We prove the soundness of our definition as well as introduce an invariant which can be used to systematically construct all such operations. We also show sufficient conditions for an operation to preserve the connectedness of the plane graph to which it is applied