scholarly journals The Maximum Number of Cycles in a Graph with Fixed Number of Edges

10.37236/8747 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Andrii Arman ◽  
Sergei Tsaturian
Keyword(s):  
Upper Bound ◽  
Fixed Number ◽  
Number Of Cycles

The main problem considered in this paper is maximizing the number of cycles in a graph with given number of edges. In 2009, Király conjectured that there is constant $c$ such that any graph with $m$ edges has at most $c(1.4)^m$ cycles. In this paper, it is shown that for sufficiently large $m$, a graph with $m$ edges has at most $(1.443)^m$ cycles. For sufficiently large $m$, examples of a graph with $m$ edges and $(1.37)^m$ cycles are presented. For a graph with given number of vertices and edges an upper bound on the maximal number of cycles is given. Also, bounds tight up to a constant are presented for the maximum number of cycles in a multigraph with given number of edges, as well as in a multigraph with given number of vertices and edges.

scholarly journals Edge Mostar Indices of Cacti Graph With Fixed Cycles

2021 ◽  
Vol 9 ◽  
Author(s):  
Farhana Yasmeen ◽  
Shehnaz Akhter ◽  
Kashif Ali ◽  
Syed Tahir Raza Rizvi
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Thermodynamic Characteristics ◽  
Chemical Compounds ◽  
Fixed Number ◽  
Common Vertex ◽  
Topological Invariants ◽  
Cactus Graph ◽  
Number Of Cycles

Topological invariants are the significant invariants that are used to study the physicochemical and thermodynamic characteristics of chemical compounds. Recently, a new bond additive invariant named the Mostar invariant has been introduced. For any connected graph ℋ, the edge Mostar invariant is described as Moe(ℋ)=∑gx∈E(ℋ)|mℋ(g)−mℋ(x)|, where mℋ(g)(or mℋ(x)) is the number of edges of ℋ lying closer to vertex g (or x) than to vertex x (or g). A graph having at most one common vertex between any two cycles is called a cactus graph. In this study, we compute the greatest edge Mostar invariant for cacti graphs with a fixed number of cycles and n vertices. Moreover, we calculate the sharp upper bound of the edge Mostar invariant for cacti graphs in ℭ(n,s), where s is the number of cycles.

scholarly journals An Upper Bound on the k-Modem Illumination Problem

2015 ◽  
Vol 25 (04) ◽  
pp. 299-308
Author(s):  
Frank Duque ◽  
Carlos Hidalgo-Toscano
Keyword(s):  
Upper Bound ◽  
Line Segment ◽  
Time Algorithm ◽  
Fixed Number ◽  
Light Sources ◽  
Wireless Devices ◽  
Orthogonal Polygon ◽  
Illumination Problem ◽  
Number Formula ◽  
Interior Points

A variation on the classical polygon illumination problem was introduced in [Aichholzer et al. EuroCG’09]. In this variant light sources are replaced by wireless devices called [Formula: see text]-modems, which can penetrate a fixed number [Formula: see text], of “walls”. A point in the interior of a polygon is “illuminated” by a [Formula: see text]-modem if the line segment joining them intersects at most [Formula: see text] edges of the polygon. It is easy to construct polygons of [Formula: see text] vertices where the number of [Formula: see text]-modems required to illuminate all interior points is [Formula: see text]. However, no non-trivial upper bound is known. In this paper we prove that the number of kmodems required to illuminate any polygon of [Formula: see text] vertices is [Formula: see text]. For the cases of illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we give a tighter bound of [Formula: see text]. Moreover, we present an [Formula: see text] time algorithm to achieve this bound.

scholarly journals EFFICIENT REGRESSIONS VIA OPTIMALLY COMBINING QUANTILE INFORMATION

2014 ◽  
Vol 30 (6) ◽  
pp. 1272-1314 ◽  
Author(s):  
Zhibiao Zhao ◽  
Zhijie Xiao
Keyword(s):  
Upper Bound ◽  
Regression Models ◽  
Statistical Estimation ◽  
Asymptotic Variance ◽  
Fixed Number ◽  
Efficient Estimation ◽  
Superior Performance ◽  
Quantile Regressions ◽  
Monte Carlo Experiments ◽  
Combining Information

We develop a generally applicable framework for constructing efficient estimators of regression models via quantile regressions. The proposed method is based on optimally combining information over multiple quantiles and can be applied to a broad range of parametric and nonparametric settings. When combining information over a fixed number of quantiles, we derive an upper bound on the distance between the efficiency of the proposed estimator and the Fisher information. As the number of quantiles increases, this upper bound decreases and the asymptotic variance of the proposed estimator approaches the Cramér–Rao lower bound under appropriate conditions. In the case of nonregular statistical estimation, the proposed estimator leads to super-efficient estimation. We illustrate the proposed method for several widely used regression models. Both asymptotic theory and Monte Carlo experiments show the superior performance over existing methods.

Aeroelastic flutter control of a bridge using rotating mass dampers and winglets

2020 ◽  
Vol 26 (23-24) ◽  
pp. 2185-2192
Author(s):  
Kamal K Bera ◽  
Naresh K Chandiramani
Keyword(s):  
Angular Speed ◽  
Fixed Number ◽  
Control Input ◽  
Vertical Force ◽  
Flat Plates ◽  
Aeroelastic Flutter ◽  
Continuous Rotation ◽  
Rational Function Approximation ◽  
Number Of Cycles ◽  
The Time Domain

Flutter control of a bridge deck section using a combination of aerodynamic and mechanical measures, that is controllable winglets and rotating mass dampers, is considered. Deck and winglets are considered as flat plates for their aerodynamics. Self-excited wind forces are represented in the time domain using the Scanlan–Tomko model with Roger’s rational function approximation for flutter derivatives. Winglet rotation relative to the deck is the control input generated by the variable-gain output feedback controller that uses vertical and torsional displacements of the deck as measured outputs. Control using winglets enhances the critical speed to twice the uncontrolled flutter speed. Further attenuation of vertical response is obtained by using rotating mass dampers configured to provide only a resultant vertical force due to counter-rotating unbalanced masses. The rotors are driven at a constant angular speed, and start–stop criteria are applied. This generates additional vertical force on the deck that is mostly out of phase with its vertical velocity. It yields better control than the damper operated in a continuous rotation mode for a fixed number of cycles. A maximum reduction of 15% in root mean square vertical response is obtained when compared with control using winglets only.

scholarly journals Locating-Total Domination Number of Cacti Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jianxin Wei ◽  
Uzma Ahmad ◽  
Saira Hameed ◽  
Javaria Hanif
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Unicyclic Graphs ◽  
Total Dominating Set ◽  
Bicyclic Graphs ◽  
Number Of Cycles

For a connected graph J, a subset W ⊆ V J is termed as a locating-total dominating set if for a ∈ V J ,   N a ∩ W ≠ ϕ , and for a ,   b ∈ V J − W ,   N a ∩ W ≠ N b ∩ W . The number of elements in a smallest such subset is termed as the locating-total domination number of J. In this paper, the locating-total domination number of unicyclic graphs and bicyclic graphs are studied and their bounds are presented. Then, by using these bounds, an upper bound for cacti graphs in terms of their order and number of cycles is estimated. Moreover, the exact values of this domination variant for some families of cacti graphs including tadpole graphs and rooted products are also determined.

scholarly journals Correlation between Thermal Behaviour of AA5754-H111 during Fatigue Loading and Fatigue Strength at Fixed Number of Cycles

Materials ◽  
2018 ◽  
Vol 11 (5) ◽  
pp. 719 ◽  
Author(s):  
Rosa De Finis ◽  
Davide Palumbo ◽  
Livia Serio ◽  
Luigi De Filippis ◽  
Umberto Galietti
Keyword(s):  
Fatigue Strength ◽  
Thermal Behaviour ◽  
Fatigue Loading ◽  
Fixed Number ◽  
Number Of Cycles

scholarly journals SYMMETRIC AND ASYMMETRIC RAMSEY PROPERTIES IN RANDOM HYPERGRAPHS

2017 ◽  
Vol 5 ◽  
Author(s):  
LUCA GUGELMANN ◽  
RAJKO NENADOV ◽  
YURY PERSON ◽  
NEMANJA ŠKORIĆ ◽  
ANGELIKA STEGER ◽  
...  
Keyword(s):  
Random Graph ◽  
Upper Bound ◽  
Natural Extension ◽  
Fixed Number ◽  
The Other ◽  
Ramsey Problem ◽  
Uniform Hypergraphs ◽  
General Bound ◽  
Random Hypergraphs ◽  
Monochromatic Copy

A celebrated result of Rödl and Ruciński states that for every graph $F$, which is not a forest of stars and paths of length 3, and fixed number of colours $r\geqslant 2$ there exist positive constants $c,C$ such that for $p\leqslant cn^{-1/m_{2}(F)}$ the probability that every colouring of the edges of the random graph $G(n,p)$ contains a monochromatic copy of $F$ is $o(1)$ (the ‘0-statement’), while for $p\geqslant Cn^{-1/m_{2}(F)}$ it is $1-o(1)$ (the ‘1-statement’). Here $m_{2}(F)$ denotes the 2-density of $F$. On the other hand, the case where $F$ is a forest of stars has a coarse threshold which is determined by the appearance of a certain small subgraph in $G(n,p)$. Recently, the natural extension of the 1-statement of this theorem to $k$-uniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, Rödl and Schacht. In particular, they showed an upper bound of order $n^{-1/m_{k}(F)}$ for the 1-statement, where $m_{k}(F)$ denotes the $k$-density of $F$. Similarly as in the graph case, it is known that the threshold for star-like hypergraphs is given by the appearance of small subgraphs. In this paper we show that another type of threshold exists if $k\geqslant 4$: there are $k$-uniform hypergraphs for which the threshold is determined by the asymmetric Ramsey problem in which a different hypergraph has to be avoided in each colour class. Along the way we obtain a general bound on the 1-statement for asymmetric Ramsey properties in random hypergraphs. This extends the work of Kohayakawa and Kreuter, and of Kohayakawa, Schacht and Spöhel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.

scholarly journals An Erdős-Ko-Rado Theorem for Permutations with Fixed Number of Cycles

10.37236/4071 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Cheng Yeaw Ku ◽  
Kok Bin Wong
Keyword(s):  
Positive Integer ◽  
Fixed Points ◽  
Fixed Number ◽  
Stirling Number ◽  
Disjoint Cycles ◽  
Set Of Permutations ◽  
Common Cycles ◽  
Number Of Cycles ◽  
Positive Integers

Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For a positive integer $k$, define $S_{n,k}$ to be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles, i.e.,\[ S_{n,k} = \{\pi \in S_{n}: \pi = c_{1}c_{2} \cdots c_{k}\},\] where $c_1,c_2,\dots ,c_k$ are disjoint cycles. The size of $S_{n,k}$ is $\left [ \begin{matrix}n\\ k \end{matrix}\right]=(-1)^{n-k}s(n,k)$, where $s(n,k)$ is the Stirling number of the first kind. A family $\mathcal{A} \subseteq S_{n,k}$ is said to be $t$-cycle-intersecting if any two elements of $\mathcal{A}$ have at least $t$ common cycles. In this paper we show that, given any positive integers $k,t$ with $k\geq t+1$, if $\mathcal{A} \subseteq S_{n,k}$ is $t$-cycle-intersecting and $n\ge n_{0}(k,t)$ where $n_{0}(k,t) = O(k^{t+2})$, then \[ |\mathcal{A}| \le \left [ \begin{matrix}n-t\\ k-t \end{matrix}\right],\]with equality if and only if $\mathcal{A}$ is the stabiliser of $t$ fixed points.

The upper bound of the number of cycles in a 2-factor of a line graph

2007 ◽  
Vol 55 (1) ◽  
pp. 72-82 ◽  
Author(s):  
Jun Fujisawa ◽  
Liming Xiong ◽  
Kiyoshi Yoshimoto ◽  
Shenggui Zhang
Keyword(s):  
Upper Bound ◽  
Line Graph ◽  
Number Of Cycles
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