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57
Approximate distance oracles
 J. ACM
"... Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in ..."
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Cited by 218 (11 self)
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Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k − 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdős, implies that Ω(n 1+1/k) 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. The most impressive feature of our data structure is its constant query time, hence the name “oracle”. Previously, data structures that used only O(n 1+1/k) space had a query time of Ω(n 1/k). Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs. 1
Extensions of finite generalized quadrangles
 In Symposia Mathematica, Vol. XXVIII
, 1983
"... 1.1 Axioms and definitions.................................... 1 1.2 Restrictions on the parameters............................... 2 1.3 Regularity, antiregularity, semiregularity, and property (H)............... 3 ..."
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Cited by 138 (12 self)
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1.1 Axioms and definitions.................................... 1 1.2 Restrictions on the parameters............................... 2 1.3 Regularity, antiregularity, semiregularity, and property (H)............... 3
Constructions for Cubic Graphs With Large Girth
 Electronic Journal of Combinatorics
, 1998
"... The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a welldefined 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 ..."
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Cited by 36 (0 self)
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The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a welldefined 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...
Spreads, translation planes and Kerdock sets. I
, 1982
"... In an orthogonal vector space of type l)/(4n, q), a spread is a family of q2nl+ totally singular 2nspaces which induces a partition of the singular points; these spreads are closely related to Kerdock sets. In a 2mdimensional vector space over GF(q), a spread is a family of q + subspaces of dim ..."
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Cited by 35 (18 self)
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In an orthogonal vector space of type l)/(4n, q), a spread is a family of q2nl+ totally singular 2nspaces which induces a partition of the singular points; these spreads are closely related to Kerdock sets. In a 2mdimensional 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.
On the Pauli graphs of Nqudits
, 2007
"... A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of Nqudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per twoqubits, all basic properties and ..."
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Cited by 13 (10 self)
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A comprehensive graph theoretical and finite geometrical study of the commutation relations between the generalized Pauli operators of Nqudits is performed in which vertices/points correspond to the operators and edges/lines join commuting pairs of them. As per twoqubits, all basic properties and partitionings of the corresponding Pauli graph are embodied in the geometry of the generalized quadrangle of order two. Here, one identifies the operators with the points of the quadrangle and groups of maximally commuting subsets of the operators with the lines of the quadrangle. The three basic partitionings are (a) a pencil of lines and a cube, (b) a Mermin’s array and a bipartitepart and (c) an independent set and the Petersen graph. These factorizations stem naturally from the existence of three distinct geometric hyperplanes of the quadrangle, namely a set of points collinear with a given point, a grid and an ovoid, which answer to three distinguished subsets of the Pauli graph, namely a set of six operators commuting with a given one, a Mermin’s square, and set of five mutually noncommuting operators, respectively. The generalized Pauli graph for multiple qubits is found to follow from symplectic polar spaces of order two, where maximal totally isotropic subspaces stand for maximal subsets of mutually commuting operators. The substructure of the (strongly regular) Nqubit Pauli graph is shown to be pseudogeometric, i. e., isomorphic to a graph of a partial geometry. Finally, the (not strongly regular) Pauli graph of a twoqutrit system is introduced, leaving open its possible link to more abstract and exotic finite geometries.
Generalized Quadrangles Associated with G_2(q)
 JOURNAL OF COMBINATORIAL THEORY, SERIES A 29,212219 (1980)
, 1980
"... 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). ..."
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Cited by 9 (2 self)
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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).
Some Exceptional 2Adic Buildings
 JOURNAL OF ALGEBRA
, 1985
"... [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], a ..."
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Cited by 9 (3 self)
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[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 lowdimensional 2adic 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 flagtransitive on the corresponding affine buildings and can be written using matrices with entries in the subring Z [4] of the rationals. (These flagtransitive groups are just Q(Z[+], f,) and the automorphism group of the nonsplit 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 flagtransitive 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 lowdimensional orthogonal groups is a reflection of the exceptional Weyl groups. Sections 35 are concerned with
Generalized quadrangles weakly embedded of degree > 2 in projective space
 Forum Math
, 1999
"... 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 ..."
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Cited by 8 (2 self)
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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 finitedimensional), 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
Pauli graph and finite projective lines/geometries
, 2007
"... The commutation relations between the generalized Pauli operators of Nqudits (i. e., N plevel quantum systems), and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometrical pattern. One may identify vertices/points with the operators so that edges/lines j ..."
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Cited by 6 (5 self)
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The commutation relations between the generalized Pauli operators of Nqudits (i. e., N plevel quantum systems), and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometrical pattern. One may identify vertices/points with the operators so that edges/lines join commuting pairs of them to form the socalled Pauli graph P p N. As per twoqubits (p = 2, N = 2) all basic properties and partitionings of this graph are embodied in the geometry of the symplectic generalized quadrangle of order two, W(2). The structure of the twoqutrit (p = 3, N = 2) graph is more involved; here it turns out more convenient to deal with its dual in order to see all the parallels with the twoqubit case and its surmised relation with the geometry of generalized quadrangle Q(4, 3), the dual of W(3). Finally, the generalized adjacency graph for multiple (N> 3) qubits/qutrits is shown to follow from symplectic polar spaces of order two/three. The relevance of these mathematical concepts to mutually unbiased bases and to quantum entanglement is also highlighted in some detail.
Polarities and 2kcyclefree graphs
 Discrete Math
, 1999
"... 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 met ..."
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Cited by 6 (1 self)
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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.