scholarly journals On the Number of Matchings in Regular Graphs

10.37236/834 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
S. Friedland ◽  
E. Krop ◽  
K. Markström
Keyword(s):  
Lower Bounds ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Expected Value ◽  
Upper And Lower Bounds ◽  
Regular Graphs

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the expected value of the number of $m$-matchings of $r$-regular bipartite graphs on $2n$ vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for $m$-matchings in $r$-regular bipartite graphs on $2n$ vertices, and their asymptotic versions for infinite $r$-regular bipartite graphs. We prove these conjectures for $2$-regular bipartite graphs and for $m$-matchings with $m\le 4$.

scholarly journals A Turán Type Problem Concerning the Powers of the Degrees of a Graph

10.37236/1525 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Degree Sequence ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Type Problem ◽  
Maximum Value ◽  
Turán Number

For a graph $G$ whose degree sequence is $d_{1},\ldots ,d_{n}$, and for a positive integer $p$, let $e_{p}(G)=\sum_{i=1}^{n}d_{i}^{p}$. For a fixed graph $H$, let $t_{p}(n,H)$ denote the maximum value of $e_{p}(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $t_{1}(n,H)$ is twice the Turán number of $H$. In this paper we consider the case $p>1$. For some graphs $H$ we obtain exact results, for some others we can obtain asymptotically tight upper and lower bounds, and many interesting cases remain open.

scholarly journals Dispersing Hash Functions

2000 ◽  
Vol 7 (36) ◽  
Author(s):  
Rasmus Pagh
Keyword(s):  
Lower Bounds ◽  
Random Function ◽  
Hash Functions ◽  
Expected Value ◽  
Independent Interest ◽  
Upper And Lower Bounds ◽  
Related Issue ◽  
Logarithmic Factor

A new hashing primitive is introduced: dispersing hash functions. A family<br />of hash functions F is dispersing if, for any set S of a certain size and random<br />h in F, the expected value of |S|−|h[S]| is not much larger than the expectancy<br />if h had been chosen at random from the set of all functions.<br />We give tight, up to a logarithmic factor, upper and lower bounds on the<br />size of dispersing families. Such families previously studied, for example <br />universal families, are significantly larger than the smallest dispersing families,<br />making them less suitable for derandomization. We present several applications<br /> of dispersing families to derandomization (fast element distinctness, set<br />inclusion, and static dictionary initialization). Also, a tight relationship <br />between dispersing families and extractors, which may be of independent interest,<br />is exhibited.<br />We also investigate the related issue of program size for hash functions<br />which are nearly perfect. In particular, we exhibit a dramatic increase in<br />program size for hash functions more dispersing than a random function.

A Note on the Multi-Colored Bipartite Ramsey Number for Paths

2021 ◽  
pp. 2150041
Author(s):  
Hanxiao Qiao ◽  
Ke Wang ◽  
Suonan Renqian ◽  
Renqingcuo
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Bipartite Graphs ◽  
Ramsey Number ◽  
Upper And Lower Bounds ◽  
Number Formula

For bipartite graphs [Formula: see text], the bipartite Ramsey number [Formula: see text] is the least positive integer [Formula: see text] so that any coloring of the edges of [Formula: see text] with [Formula: see text] colors will result in a copy of [Formula: see text] in the [Formula: see text]th color for some [Formula: see text]. In this paper, we get the exact value of [Formula: see text], and obtain the upper and lower bounds of [Formula: see text], where [Formula: see text] denotes a path with [Formula: see text] vertices.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

scholarly journals Isoperimetric Numbers of Regular Graphs of High Degree with Applications to Arithmetic Riemann Surfaces

10.37236/651 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Dominic Lanphier ◽  
Jason Rosenhouse
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Riemann Surfaces ◽  
Upper And Lower Bounds ◽  
Regular Graphs ◽  
Eigenvalue Spectrum ◽  
Asymptotic Bounds ◽  
High Degree

We derive upper and lower bounds on the isoperimetric numbers and bisection widths of a large class of regular graphs of high degree. Our methods are combinatorial and do not require a knowledge of the eigenvalue spectrum. We apply these bounds to random regular graphs of high degree and the Platonic graphs over the rings $\mathbb{Z}_n$. In the latter case we show that these graphs are generally non-Ramanujan for composite $n$ and we also give sharp asymptotic bounds for the isoperimetric numbers. We conclude by giving bounds on the Cheeger constants of arithmetic Riemann surfaces. For a large class of these surfaces these bounds are an improvement over the known asymptotic bounds.

scholarly journals Counting Proper Colourings in 4-Regular Graphs via the Potts Model

10.37236/7743 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Ewan Davies
Keyword(s):  
Partition Function ◽  
Internal Energy ◽  
Lower Bounds ◽  
Regular Graph ◽  
Potts Model ◽  
Temperature Limit ◽  
Upper And Lower Bounds ◽  
Regular Graphs ◽  
Zero Temperature ◽  
First Case

We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$-state Potts model on $4$-regular graphs, for $q\ge 5$. This proves the first case of a conjecture of the author, Perkins, Jenssen and Roberts, and implies tight bounds on the antiferromagnetic Potts partition function. The zero-temperature limit gives upper and lower bounds on the number of proper $q$-colourings of $4$-regular graphs, which almost proves the case $d=4$ of a conjecture of Galvin and Tetali. For any $q \ge 5$ we prove that the number of proper $q$-colourings of a $4$-regular graph is maximised by a union of $K_{4,4}$'s.  

scholarly journals Regularity and projective dimension of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs

2019 ◽  
Vol 19 (12) ◽  
pp. 2050233 ◽  
Author(s):  
Guangjun Zhu ◽  
Li Xu ◽  
Hong Wang ◽  
Jiaqi Zhang
Keyword(s):  
Lower Bounds ◽  
Disjoint Union ◽  
Projective Dimension ◽  
Exact Formula ◽  
Bipartite Graphs ◽  
Upper And Lower Bounds ◽  
Edge Ideal ◽  
Undirected Graphs

In this paper, we provide some precise formulas for regularity of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs. For the projective dimension of such an edge ideal, we give its exact formula. Meanwhile, we also give the upper and lower bounds of projective dimension of higher powers of such an edge ideal. As an application, we present regularity and projective dimension of powers of edge ideal of some gap-free bipartite undirected graphs. Some examples show that these formulas are related to direction selection.

Vertex-Neighbor-Scattering Number of Bipartite Graphs

2016 ◽  
Vol 27 (04) ◽  
pp. 501-509
Author(s):  
Zongtian Wei ◽  
Nannan Qi ◽  
Xiaokui Yue
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Bipartite Graphs ◽  
Upper And Lower Bounds ◽  
Number Of Components ◽  
Np Completeness ◽  
Vertex Set ◽  
Np Complete

Let G be a connected graph. A set of vertices [Formula: see text] is called subverted from G if each of the vertices in S and the neighbor of S in G are deleted from G. By G/S we denote the survival subgraph that remains after S is subverted from G. A vertex set S is called a cut-strategy of G if G/S is disconnected, a clique, or ø. The vertex-neighbor-scattering number of G is defined by [Formula: see text], where S is any cut-strategy of G, and ø(G/S) is the number of components of G/S. It is known that this parameter can be used to measure the vulnerability of spy networks and the computing problem of the parameter is NP-complete. In this paper, we discuss the vertex-neighbor-scattering number of bipartite graphs. The NP-completeness of the computing problem of this parameter is proven, and some upper and lower bounds of the parameter are also given.

THE STOCHASTIC WAVE EQUATION DRIVEN BY FRACTIONAL BROWNIAN NOISE AND TEMPORALLY CORRELATED SMOOTH NOISE

2005 ◽  
Vol 05 (01) ◽  
pp. 45-64 ◽  
Author(s):  
PETER CAITHAMER
Keyword(s):  
Wave Equation ◽  
Lower Bounds ◽  
Spatial Dimension ◽  
Expected Value ◽  
Upper And Lower Bounds ◽  
Small Deviations ◽  
Stochastic Wave Equation ◽  
Fractional Brownian Noise ◽  
Temporally Correlated ◽  
Fractional Noises

The stochastic wave equation in one spatial dimension driven by a class of fractional noises or, alternately, by a class of smooth noises with arbitrary temporal covariance is studied. In either case, the wave equation is explicitly solved and the upper and lower bounds on both the large and small deviations of several sup norms associated with the solution are given. Finally the energy of a system governed by such an equation is calculated and its expected value is found.

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