A characterization of the components of the graphs D(k,q)

Lazebnik, Felix; Ustimenko, Vasiliy A.; Woldar, Andrew J.

doi:10.1016/s0012-365x(96)83019-6

Cited by 45 publications

(37 citation statements)

References 5 publications

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“…It is the family of graphs CD(n, q) which provides the lower bound in the above inequality, being a slight improvement of the previous best lower bound Ω(ν 1+ 2 3k+3 ) given by the family of Ramanujan graphs constructed by Margulis [35], and independently by Lubotzky, Phillips and Sarnak [34]. In [28], Lazebnik, Ustimenko and Woldar proved that for all n ≥ 6 and q odd, the order of CD(n, q) is equal to 2q n− n+2 4 +1 , hence another family is needed if the lower bound in 4.1 is to be improved. For n = 2, 3, 5, the magnitude ν 1+ 1 n in the upper bound of 4.1 is attained by D(n, q) (n = 2, 3 and q odd) and by the regular generalized hexagon (n = 5).…”

Section: Examples and Applicationsmentioning

confidence: 80%

General properties of some families of graphs defined by systems of equations

Lazebnik

Woldar

2001

Journal of Graph Theory

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Abstract. In this paper we present a simple method for constructing infinite families of graphs defined by a class of systems of equations over commutative rings. We show that the graphs in all such families possess some general properties including regularity and bi-regularity, existence of special vertex colorings, and existence of covering maps -hence, embedded spectra -between every two members of the same family. Another general property, recently discovered, is that nearly every graph constructed in this manner edge-decomposes either the complete, or complete bipartite, graph which it spans.In many instances, specializations of these constructions have proved useful in various graph theory problems, but especially in many extremal problems. A short survey of the related results is included. We also show that the edgedecomposition property allows one to improve existing lower bounds for some multicolor Ramsey numbers.

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Section: Examples and Applicationsmentioning

confidence: 80%

General properties of some families of graphs defined by systems of equations

Lazebnik

Woldar

2001

Journal of Graph Theory

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“…We get, therefore, the following interesting corollary: 10 Ω(n 6/5 ) , O(n 5/4 ) [Tit59], [Ben66], [LU93] 12 Θ(n 6/5 ) [Tit59], [Ben66], [Wen91], [LU93] 14 Ω(n 9/8 ) , O(n 7/6 ) [LUW95], [LUW96] 16 Ω(n 10/9 ) , O(n 8/7 ) [WU93], [LUW95] 4r , r ≥ 5 Ω(n…”

Section: Sparse Spanners and Tree Coversmentioning

confidence: 81%

Approximate distance oracles

Thorup¹,

Zwick²

2001

Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing

377

752

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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.

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“…To distinguish the points from the lines we use parentheses and brackets: If x ∈ V , then (x) ∈ P and [x] ∈ L. It will also be advantageous to adopt the notation for co-ordinates of points and lines introduced in [17] for the case of general commutative ring K: p 0,1 , p 1,1 , p 1,2 , p 2,1 , p 2,2 , p 2,2 , p 2,3 , . .…”

Section: The Incidence Structures Defined Over Commutative Ringsmentioning

confidence: 99%

“…For K = F q the following statement had been formulated in [17]. Let k 6, t = [(k + 2)/4], and let u = (u α , u 11 , · · · , u tt , u tt , u t,t+1 , u t+1,t , · · ·) be a vertex of D(k, K) (α ∈ {(1, 0), (0, 1)}.…”

Section: The Incidence Structures Defined Over Commutative Ringsmentioning

confidence: 99%

See 1 more Smart Citation

On the implementation of cryptoalgorithms based on algebraic graphs over some commutative rings

Kotorowicz¹,

Ustimenko²

2008

Condens. Matter Phys.

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The paper is devoted to computer implementation of some graph based stream ciphers. We compare the time performance of this new algorithm with fast, but no very secure RC4, and with DES. It turns out that some of new algorithms are faster than RC4. They satisfy the Madryga requirements, which is unusual for stream ciphers (like RC4). The software package with new encryption algorithms is ready for the demonstration.

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A characterization of the components of the graphs D(k,q)

Cited by 45 publications

References 5 publications

General properties of some families of graphs defined by systems of equations

General properties of some families of graphs defined by systems of equations

Approximate distance oracles

On the implementation of cryptoalgorithms based on algebraic graphs over some commutative rings

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