Let G = (V; E) be an undirected weighted graph with jV j = n and jEj = m. Let k 1 be an integer. We show that G = (V; E) can be preprocessed in O(kmn ) expected time, constructing a data structure of size O(kn ), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k \Gamma 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k \Gamma 1. We show that a 1963 girth conjecture of Erdos, implies ) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal.

In an orthogonal vector space of type l)/(4n, q), a spread is a family of q2n-l+ totally singular 2n-spaces which induces a partition of the singular points; these spreads are closely related to Kerdock sets. In a 2m-dimensional vector space over GF(q), a spread is a family of q + subspaces of dimension m which induces a partition of the points of the underlying projective space; these spreads correspond to affine translation planes. By combining geometric, group theoretic and matrix methods, new types of spreads are constructed and old examples are studied. New Kerdock sets and new translation planes are obtained having various interesting properties.

The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a well-defined integer ¯ 0 (g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value ¯ 0 (g) is attained by a graph whose girth is exactly g. The values of ¯ 0 (g) when 3 g 8 have been known for over thirty years. For these values of g each minimal graph is unique and, apart from the case g = 7, a simple lower bound is attained. This paper is mainly concerned with what happens when g 9, where the situation is quite different. Here it is known that the simple lower bound is attained if and only if g = 12. A number of techniques are described, with emphasis on the construction of families of graphs fG i g for which the number of vertices n i and the girth g i are such that n i 2 cg i for some finite constant c. The optimum value of c is known to lie between 0:5 and 0:75. At the end of the p...

[19] has initiated the study of geometries having properties strongly resembling those of buildings (compare [6]). His main result is that, in general, such a geometry is the image of a building under a suitable type of morphism. These geometries that are almost buildings were called GABS in [lo], and finite examples were described in [2, 8, 10, 15, 161 that are not buildings but have highly transitive groups. In this paper we will proceed in the opposite direction. An unexpected type of description is given for the afftne buildings associated to some low-dimensional 2-adic orthogonal groups. This description makes it easy to produce infinitely many finite GABS having large groups. Specifically, we construct subgroups of 0’(8, a,), O(7, a,), LV(6, Q,) and G2(U4J that are flag-transitive on the corresponding affine buildings and can be written using matrices with entries in the subring Z [4] of the rationals. (These flag-transitive groups are just Q(Z[+], f,) and the automorphism group of the non-split Cayley algebra over Z [f], where k = 8, 7 or 6 and fk is the quadratic form C: xi.) If m is any odd integer> 1 and these matrices are viewed modulo m then a finite GAB is obtained having a flag-transitive group and one of the following diagrams. Each connected rank 2 subdiagram corresponds to PSL(3, 2), Sp(4,2) or G,(2). This exceptional behavior of low-dimensional orthogonal groups is a reflection of the exceptional Weyl groups. Sections 3-5 are concerned with

If q 3 2 (mod 3), a generalized quadrangle with parameters q, q2 is constructed from the generalized hexagon associated with the group G_2(q).

Abstract. Motivated by the incidence problems between points and flats of a symplectic polar space, we study a large class of submodules of the space of functions on the standard module of a finite symplectic group of odd characteristic. Our structure results on this class of submodules allow us to determine the p-ranks of the incidence matrices between points and flats of the symplectic polar space. In particular, we give an explicit formula for the p-rank of the incidence matrix between the points and lines of the symplectic generalized quadrangle W(3, q), where q is an odd prime power. Combined with the earlier results of Sastry and Sin on the 2-rank of W(3, 2 t), it completes the determination of the p-ranks of W(3, q). 1.

In this paper, we classify all generalized quadrangles weakly embedded of degree 2 in projective space. More exactly, given a (possibly infinite) generalized quadrangle Γ = (P, L, I) and a map π from P (respectively L) to the set of points (respectively lines) of a projective space PG(V), V a vector space over some skew field (not necessarily finite-dimensional), such that: (i) π is injective on points, (ii) if x ∈Pand L ∈Lwith xIL, then x π is incident with L π in PG(V), (iii) the set of points {xπ | x ∈P}generates PG(V), (iv) if x, y ∈ P such that yπ is contained in the subspace of PG(V) generated by the set {z π | z is collinear with x in Γ}, then y is collinear with x in Γ, (v) there exists a line of PG(V) not in the image of π and

Currently, there are no known explicit algorithms for the great majority of graph problems in the dynamic distributed message-passing model. Instead, most state-of-the-art dynamic distributed algorithms are constructed by composing a static algorithm for the problem at hand with a simulation technique that converts static algorithms to dynamic ones. We argue that this powerful methodology does not provide satisfactory solutions for many important dynamic distributed problems, and this necessitates developing algorithms for these problems from scratch. In this paper we develop a fully dynamic distributed algorithm for maintaining sparse spanners. Our algorithm improves drastically the quiescence time of the state-of-the-art algorithm for the problem. Moreover, we show that the quiescence time of our algorithm is optimal up to a small constant factor. In addition, our algorithm improves significantly upon the state-of-the-art algorithm in all efficiency parameters, specifically, it has smaller quiescence message and space complexities, and smaller local processing time. Finally, our algorithm is self-contained and fairly simple, and is, consequently, amenable to implementation on unsophisticated network devices.

Let C2k be the cycle on 2k vertices, and let ex(v, C2k) denote the greatest number of edges in a simple graph on v vertices which contains no subgraph isomorphic to C2k. In this paper we discuss a method which allows one to sometimes improve numerical constants in lower bounds for ex(v,C2k). The method utilizes polarities in certain rank two geometries. It is applied to refute some conjectures about the values of ex(v,C2k), and to construct some new examples of graphs having certain restrictions on the lengths of their cycles. In particular, we construct an infinite family {Gi} of C6–free graphs with |E(Gi) | ∼ 1 2 |V (Gi) | 4/3, i → ∞, which improves the constant in the previous best lower bound on ex(v,C6) from 2/3 4/3 ≈.462 to 1/2.

Abstract. Let K/Q be a number field with euclidean ring of integers O. Let Γ be a finite-index torsion-free subgroup of the symplectic group Sp2n(O), and let N be the cohomological dimension of Γ. We exhibit a finite, geometrically-defined spanning set of H N (Γ; Z) by generalizing the modular symbol algorithm of Ash and Rudolph for SLn(O). 1.