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.

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 Hereditary quasirandomness without regularity

2017 ◽  
Vol 164 (3) ◽  
pp. 385-399 ◽  
Author(s):  
DAVID CONLON ◽  
JACOB FOX ◽  
BENNY SUDAKOV
Keyword(s):  
Alternative Proof ◽  
Regularity Lemma ◽  
Original Proof ◽  
Vertex Graph

AbstractA result of Simonovits and Sós states that for any fixed graph H and any ε > 0 there exists δ > 0 such that if G is an n-vertex graph with the property that every S ⊆ V(G) contains pe(H) |S|v(H) ± δ nv(H) labelled copies of H, then G is quasirandom in the sense that every S ⊆ V(G) contains $\frac{1}{2}$p|S|2± ε n2 edges. The original proof of this result makes heavy use of the regularity lemma, resulting in a bound on δ−1 which is a tower of twos of height polynomial in ε−1. We give an alternative proof of this theorem which avoids the regularity lemma and shows that δ may be taken to be linear in ε when H is a clique and polynomial in ε for general H. This answers a problem raised by Simonovits and Sós.

Mechanical Structural Interference Detection for Dedicated Hybrid Transmissions

2021 ◽  
Vol 143 (9) ◽  
Author(s):  
Hanqiao Sun ◽  
Xiangyang Xu ◽  
Yanfang Liu ◽  
Peng Dong ◽  
Shuhan Wang ◽  
...  
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Planetary Gear ◽  
Modeling Method ◽  
Structural Synthesis ◽  
Original Graph ◽  
Power Sources ◽  
Detection Approach ◽  
Hybrid Transmission ◽  
Planetary Gear System

Abstract Planetary gear set (PGS) has been one of the best components to constitute a transmission configuration, including the dedicated hybrid transmission (DHT). Using different synthesis approaches, the DHT configurations can be obtained through algorithms. However, different synthesis results correspond to different connection states of the planetary gear system. There are a certain number of results that violate the motion requirements of the mechanical principal need to be detected and removed. Therefore, this paper presents a novel modeling method to systematically remove the interference structures, with graph theory in structural synthesis. Based on the original graph theory, this paper proposes an equivalent replacement modeling method to convert the motor graph model into a brake-like graph model. Based on the conversion, avoid the appearance of the hanging points in the graph model. By applying the proposed approach, a DHT structure proves the feasibility of the method. The proposed detection approach can systematically detect all the PGS-based transmission with multi-PGSs, multi-shifting elements, and multi-power sources.

Entanglement of formation and concurrence

2001 ◽  
Vol 1 (1) ◽  
pp. 27-44
Author(s):  
William K. Wootters
Keyword(s):  
Current Understanding ◽  
Related Concept ◽  
Explicit Formulas ◽  
Entanglement Of Formation ◽  
Bipartite States

This paper reviews our current understanding of entanglement of formation and the related concept of concurrence, including discussions of additivity, the problem of finding explicit formulas, and connections between concurrence and other propertis of bipartite states.

scholarly journals Zero-one completely positive matrices and the A(R, S) classes

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
G. Dahl ◽  
T. A. Haufmann
Keyword(s):  
Related Concept ◽  
Completely Positive ◽  
Completely Positive Matrices ◽  
Positive Matrices

AbstractA matrix of the form A = BBT where B is nonnegative is called completely positive (CP). Berman and Xu (2005) investigated a subclass of CP-matrices, called f0, 1g-completely positive matrices. We introduce a related concept and show connections between the two notions. An important relation to the so-called cut cone is established. Some results are shown for f0, 1g-completely positive matrices with given graphs, and for {0,1}-completely positive matrices constructed from the classes of (0, 1)-matrices with fixed row and column sums.

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