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Inspired by connections described in a recent paper by Mark L. Lewis, between the common divisor graph Γ(X) and the prime vertex graph Δ(X), for a set X of positive integers, we define the bipartite divisor graph B(X), and show that many of these connections flow naturally from properties of B(X). In particular we establish links between parameters of these three graphs, such as number and diameter of components, and
we characterise bipartite graphs that can arise as B(X) for some X. Also we obtain necessary and sufficient conditions, in terms of subconfigurations of B(X), for one of Γ(X) or Δ(X) to contain a complete subgraph of size 3 or 4. 相似文献
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Edwin R. van Dam 《Journal of Algebraic Combinatorics》1998,7(3):321-332
We give a bound on the sizes of two sets of vertices at a given minimum distance in a graph in terms of polynomials and the Laplace spectrum of the graph. We obtain explicit bounds on the number of vertices at maximal distance and distance two from a given vertex, and on the size of two equally large sets at maximal distance. For graphs with four eigenvalues we find bounds on the number of vertices that are not adjacent to a given vertex and that have µ common neighbours with that vertex. Furthermore we find that the regular graphs for which the bounds are tight come from association schemes. 相似文献
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借助于拓扑加群∑2的齐性,本给出了如下定理的一个证明:任一个图G上的任一既无孤立点也无内点的非空闭子集均是个Cantor集。 相似文献
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Imre Z. Ruzsa 《The Ramanujan Journal》2005,9(1-2):77-82
We estimate the maximal number of elements that can be selected from a set of n numbers so that the sum of two selected elements are never in the original set. We improve Choi’s upper bound of n2/5 to ec√log n n.To Professor Nicolas, on the occasion of his 60th birthday.2000 Mathematics Subject Classification: Primary—11B75Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 38396, T43623 and T42750. 相似文献
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Sanming Zhou 《Monatshefte für Mathematik》2003,139(1):69-81
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 相似文献
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We study almost convex subsets of spaces with one-sided curvature bounds. We derive some characterisations and properties
of sets of positive reach in Riemannian manifolds. 相似文献
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Several rules known to be valid within the set theory are revisited and considered from the point of view of the question whether they remain satisfied when sets are replaced with subspaces from Cn,1. It is shown that some of the set relationships hold true upon such a replacement, however, there is also an extensive class of relationships whose validity is no longer guaranteed; in this class the commutativity of the orthogonal projectors onto the involved subspaces proves to play an important role. Some related results inspired by probability theory are established as well. 相似文献
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Isosceles Planar Subsets 总被引:2,自引:0,他引:2
P. Fishburn 《Discrete and Computational Geometry》1998,19(3):391-398
A finite planar set is k -isosceles for k ≥ 3 if every k -point subset of the set contains a point equidistant from two others. There are three nonsimilar 3-isosceles sets with five
points and one with six points. Eleven 4-isosceles sets with eight points are noted, and it is conjectured that no 4-isosceles
set has nine points. Exactly one 4-isosceles 8-set has four points on a line, and every 4-isosceles set that includes the
vertices of a square has fewer than eight points.
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<onlinepub>26 June, 1998
<editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt;
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Received February 1, 1997, and in revised form June 24, 1997. 相似文献
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Four distinct elements a, b, c, and d of a poset form a diamond if \(a< b and \(a . A subset of a poset is diamond-free if no four elements of the subset form a diamond. Even in the Boolean lattices, finding the size of the largest diamond-free subset remains an open problem. In this paper, we consider the linear lattices—poset of subspaces of a finite dimensional vector space over a finite field of order q—and extend the results of Griggs et al. (J. Combin. Theory Ser. A 119(2):310–322, 2012) on the Boolean lattices, to prove that the number of elements of a diamond-free subset of a linear lattice can be no larger than \(2+\frac {1}{q+1}\) times the width of the lattice, so that this fraction tends to 2 as \(q \longrightarrow \infty \) . In addition, using an algebraic technique, we introduce so-called diamond matchings, and prove that for linear lattices of dimensions up to 5, the size of a largest diamond-free subset is equal to the sum of the largest two rank numbers of the lattice. 相似文献
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王继顺 《数学的实践与认识》2017,(7):152-160
通过揭示完全蛛网图和渔网图的结构特点,研究了它们的邻点可区别I-全染色问题,并运用构造法给出了其邻点可区别I-全染色,从而获得了它们的邻点可区别I-全色数. 相似文献
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Ts. Ch.-D. Batueva 《Journal of Mathematical Sciences》2013,195(6):771-778