scholarly journals Matrix-Free Proof of a Regularity Characterization

10.37236/1732 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
A. Czygrinow ◽  
B. Nagle
Keyword(s):  
Simple Proof ◽  
Regularity Lemma ◽  
Central Concept ◽  
Matrix Argument ◽  
The Matrix ◽  
Szemeredi's Regularity Lemma ◽  
Matrix Free ◽  
Regular Pair

The central concept in Szemerédi's powerful regularity lemma is the so-called $\epsilon$-regular pair. A useful statement of Alon et al. essentially equates the notion of an $\epsilon$-regular pair with degree uniformity of vertices and pairs of vertices. The known proof of this characterization uses a clever matrix argument. This paper gives a simple proof of the characterization without appealing to the matrix argument of Alon et al. We show the $\epsilon$-regular characterization follows from an application of Szemerédi's regularity lemma itself.

scholarly journals Tilings in Randomly Perturbed Dense Graphs

2018 ◽  
Vol 28 (2) ◽  
pp. 159-176 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
ANDREW TREGLOWN ◽  
ADAM ZSOLT WAGNER
Keyword(s):  
High Probability ◽  
Minimum Degree ◽  
Graph Model ◽  
Regularity Lemma ◽  
Arbitrary Graph ◽  
Random Graph Model ◽  
Dense Graphs ◽  
Szemeredi's Regularity Lemma ◽  
Vertex Disjoint ◽  
Special Case

A perfect H-tiling in a graph G is a collection of vertex-disjoint copies of a graph H in G that together cover all the vertices in G. In this paper we investigate perfect H-tilings in a random graph model introduced by Bohman, Frieze and Martin [6] in which one starts with a dense graph and then adds m random edges to it. Specifically, for any fixed graph H, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect H-tiling with high probability. Our proof utilizes Szemerédi's Regularity Lemma [29] as well as a special case of a result of Komlós [18] concerning almost perfect H-tilings in dense graphs.

scholarly journals Absorption and Photoluminescence Study of Cds Quantum Dots: The Role of Host Matrix and Nanocrystal Size and Density

1999 ◽  
Vol 571 ◽  
Author(s):  
Yu.P. Rakovich ◽  
A.G. Rolo ◽  
M.V. Stepikhova ◽  
M.I. Vasilevskiy ◽  
M.J.M. Gomes ◽  
...  
Keyword(s):  
Volume Fraction ◽  
Optical Transition ◽  
Spectral Position ◽  
Host Matrix ◽  
The Matrix ◽  
Mean Size ◽  
Free Films ◽  
Matrix Free ◽  
Cds Crystallites

ABSTRACTIn this paper we present results of the absorption and photoluminescence (PL) of CdSdoped Si02 films fabricated by RF co-sputtering (semiconductor volume fraction f=1–15%, nano-crystallite's mean size 5–7nm) and matrix-free films of close-packed CdS nanocrystallites (f∼30%, size 2–5nm) produced by an original chemical method. The absorption spectra have been modelled using the modified Maxwell-Garnett model. This gives the e-h pair state energies and evidence of a strong absorption in the glass matrix containing CdS. The temperature dependence of the spectral position and broadening of the PL peak is analysed. It is concluded that a photo-generated hole is captured on an acceptor-type trap before the radiative recombination with a confined electron. The excitation of this ‘band-edge’ PL occurs through some states in the matrix and directly in the CdS crystallites for the two kinds of samples, respectively. The temperature coefficients of the optical transition energies for the nearly matrix-free films are similar to those of bulk CdS, while for the CdS/glass films they are smaller. This may be because of the different boundary conditions for the thermal expansion of CdS crystallites.

scholarly journals Stability for Vertex Cycle Covers

10.37236/5185 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
József Balogh ◽  
Frank Mousset ◽  
Jozef Skokan
Keyword(s):  
Minimum Degree ◽  
Independent Set ◽  
Stability Result ◽  
Natural Generalization ◽  
Regularity Lemma ◽  
Property A ◽  
Vertex Graph ◽  
Szemeredi's Regularity Lemma ◽  
Cycle Covers ◽  
Positive Integers

In 1996 Kouider and Lonc proved the following natural generalization of Dirac's Theorem: for any integer $k\geq 2$, if $G$ is an $n$-vertex graph with minimum degree at least $n/k$, then there are $k-1$ cycles in $G$ that together cover all the vertices.This is tight in the sense that there are $n$-vertex graphs that have minimum degree $n/k-1$ and that do not contain $k-1$ cycles with this property. A concrete example is given by $I_{n,k} = K_n\setminus K_{(k-1)n/k+1}$ (an edge-maximal graph on $n$ vertices with an independent set of size $(k-1)n/k+1$). This graph has minimum degree $n/k-1$ and cannot be covered with fewer than $k$ cycles. More generally, given positive integers $k_1,\dotsc,k_r$ summing to $k$, the disjoint union $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$ is an $n$-vertex graph with the same properties.In this paper, we show that there are no extremal examples that differ substantially from the ones given by this construction. More precisely, we obtain the following stability result: if a graph $G$ has $n$ vertices and minimum degree nearly $n/k$, then it either contains $k-1$ cycles covering all vertices, or else it must be close (in ‘edit distance') to a subgraph of $I_{k_1n/k,k_1}+ \dotsb + I_{k_rn/k,k_r}$, for some sequence $k_1,\dotsc,k_r$ of positive integers that sum to $k$.Our proof uses Szemerédi's Regularity Lemma and the related machinery.

scholarly journals Holes in Graphs

10.37236/1618 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Yuejian Peng ◽  
Vojtech Rödl ◽  
Andrzej Ruciński
Keyword(s):  
Regularity Lemma ◽  
Related Concept ◽  
Original Graph ◽  
Large Graph ◽  
Regular Pair

The celebrated Regularity Lemma of Szemerédi asserts that every sufficiently large graph $G$ can be partitioned in such a way that most pairs of the partition sets span $\epsilon$-regular subgraphs. In applications, however, the graph $G$ has to be dense and the partition sets are typically very small. If only one $\epsilon$-regular pair is needed, a much bigger one can be found, even if the original graph is sparse. In this paper we show that every graph with density $d$ contains a large, relatively dense $\epsilon$-regular pair. We mainly focus on a related concept of an $(\epsilon,\sigma)$-dense pair, for which our bound is, up to a constant, best possible.

The RESPECT method. Simple proof of finding estimates from below for minimal singular eigenvalues of random matrices whose entries have zero means and bounded variances

2019 ◽  
Vol 27 (3) ◽  
pp. 167-175
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Lower Bounds ◽  
Simple Proof ◽  
New Method ◽  
The Matrix ◽  
Eigenvalues Of Random Matrices

Abstract The lower bounds for the minimal singular eigenvalue of the matrix whose entries have zero means and bounded variances are obtained. The new method is based on the G-method of perpendiculars and the RESPECT method.

scholarly journals NEW SOLVABLE MATRIX INTEGRALS

2004 ◽  
Vol 19 (supp02) ◽  
pp. 276-293 ◽  
Author(s):  
A. YU. ORLOV
Keyword(s):  
Complex Matrices ◽  
Tau Function ◽  
Tau Functions ◽  
Matrix Argument ◽  
Matrix Integrals ◽  
The Matrix

We generalize the Harish-Chandra-Itzykson-Zuber and certain other integrals (the Gross-Witten integral, the integrals over complex matrices and the integrals over rectangle matrices) using a notion of the tau function of the matrix argument. In this case one can reduce multi-matrix integrals to integrals over eigenvalues, which in turn are certain tau functions. We also consider a generalization of the Kontsevich integral.

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