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On a Conjecture about Degree Deviation Measure of Graphs 关于图度偏差测度的一个猜想
Pub Date : 2020-02-20 DOI: 10.22108/TOC.2020.121737.1709
A. Ghalavand, A. Ashrafi
Let G be an n-vertex graph with m edges. The degree deviation measure of G is defined as s(G)=sum v in V(G)|degG(v)-(2m/n)|, where n and m are the number of vertices and edges of G, respectively. The aim of this paper is to prove the Conjecture 4.2 of [J A de Oliveira, C S Oliveira, C Justel and N M Maia de Abreu, Measures of irregularity of graphs, Pesq. Oper. 33 (3) (2013) 383-398]. The degree deviation measure of chemical graphs under some conditions on the cyclomatic number is also computed.
设G是一个有m条边的n顶点图。G的度偏差度量定义为s(G)=sum v in v (G)|degG(v)-(2m/n)|,其中n为G的顶点数,m为G的边数。本文的目的是证明J A de Oliveira, C S Oliveira, C Justel和N M Maia de Abreu的猜想4.2,图的不规则性度量,Pesq。卷33(3)(2013)383-398]。计算了化学图在一定条件下对圈数的度偏差测度。
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引用次数: 3
Separating Bohr denseness from measurable recurrence 从可测递归中分离玻尔密度
Pub Date : 2020-02-17 DOI: 10.19086/da.26859
John T. Griesmer
We prove that there is a set of integers $A$ having positive upper Banach density whose difference set $A-A:={a-b:a,bin A}$ does not contain a Bohr neighborhood of any integer, answering a question asked by Bergelson, Hegyv'ari, Ruzsa, and the author, in various combinations. In the language of dynamical systems, this result shows that there is a set of integers $S$ which is dense in the Bohr topology of $mathbb Z$ and which is not a set of measurable recurrence. Our proof yields the following stronger result: if $Ssubseteq mathbb Z$ is dense in the Bohr topology of $mathbb Z$, then there is a set $S'subseteq S$ such that $S'$ is dense in the Bohr topology of $mathbb Z$ and for all $min mathbb Z,$ the set $(S'-m)setminus {0}$ is not a set of measurable recurrence.
我们证明了存在一个具有正上巴纳赫密度的整数集$ a $,其差集$ a- a:={a-b:a,b在a }$中不包含任何整数的玻尔邻域,回答了Bergelson、Hegyv ari、Ruzsa等人在各种组合下提出的问题。在动力系统语言中,这一结果表明在玻尔拓扑$mathbb Z$中存在一个整数集$S$,它是密集的,并且不是一个可测量的递归集。我们的证明得到以下更强的结果:如果$Ssubseteq mathbb Z$在$mathbb Z$的玻尔拓扑中是密集的,那么存在一个集合$S'subseteq S$使得$S'$在$mathbb Z$的玻尔拓扑中是密集的,并且对于所有$m mathbb Z$, $集合$(S'-m)setminus {0}$不是一个可测量的递归集。
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引用次数: 6
Demazure crystals for Kohnert polynomials Kohnert多项式的形变晶体
Pub Date : 2020-02-17 DOI: 10.1090/tran/8560
Sami H. Assaf
Kohnert polynomials are polynomials indexed by unit cell diagrams in the first quadrant defined earlier by the author and Searles that give a common generalization of Schubert polynomials and Demazure characters for the general linear group. Demazure crystals are certain truncations of normal crystals whose characters are Demazure characters. For each diagram satisfying a southwest condition, we construct a Demazure crystal whose character is the Kohnert polynomial for the given diagram, resolving an earlier conjecture of the author and Searles that these polynomials expand nonnegatively into Demazure characters. We give explicit formulas for the expansions with applications including a characterization of those diagrams for which the corresponding Kohnert polynomial is a single Demazure character.
Kohnert多项式是由作者和Searles先前定义的第一象限的单位胞图索引的多项式,它给出了一般线性群的Schubert多项式和Demazure特征的一般推广。变形晶体是正常晶体的某些截短部分,其特征为变形特征。对于每个满足西南条件的图,我们构造了一个Demazure晶体,其特征是给定图的Kohnert多项式,解决了作者和Searles早先的一个猜想,即这些多项式非负地展开为Demazure特征。我们给出了展开式的显式公式,这些展开式的应用包括相应的Kohnert多项式是单个Demazure特征的图的表征。
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引用次数: 8
Symmetrizable integer matrices having all their eigenvalues in the interval $[-2,2]$ 具有所有特征值在区间$[-2,2]$内的可对称整数矩阵
Pub Date : 2020-02-14 DOI: 10.5802/alco.113
J. McKee, C. Smyth
The adjacency matrices of graphs form a special subset of the set of all integer symmetric matrices. The description of which graphs have all their eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most 2) has been known for several decades. In 2007 we extended this classification to arbitrary integer symmetric matrices. In this paper we turn our attention to symmetrizable matrices. We classify the connected nonsymmetric but symmetrizable matrices which have entries in $Z$ that are maximal with respect to having all their eigenvalues in [-2,2]. This includes a spectral characterisation of the affine and finite Dynkin diagrams that are not simply laced (much as the graph result gives a spectral characterisation of the simply laced ones).
图的邻接矩阵是所有整数对称矩阵集合的一个特殊子集。关于哪些图的所有特征值都在区间[-2,2](即谱半径最多为2的图)的描述已经有几十年的历史了。2007年,我们将这种分类扩展到任意整数对称矩阵。在本文中,我们将注意力转向可对称矩阵。我们对连通的非对称但可对称的矩阵进行分类,这些矩阵的元素在$Z$中,它们的所有特征值在[-2,2]中是极大的。这包括仿射图和有限Dynkin图的光谱特征,而不是简单地加了条纹(就像图形结果给出了简单加了条纹的光谱特征一样)。
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引用次数: 4
Arithmetic combinatorics on Vinogradov systems 维诺格拉多夫系统的算术组合
Pub Date : 2020-01-22 DOI: 10.1090/tran/8121
Akshat Mudgal
In this paper, we present a variant of the Balog-Szemeredi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon.
本文给出了Vinogradov系统的balog - szemeredii - gowers定理的一个变体。然后,我们用我们的结果来推导出和积现象的更高程度的类比。
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引用次数: 5
Complex Hadamard diagonalisable graphs 复哈达玛可对角图
Pub Date : 2020-01-01 DOI: 10.1016/j.laa.2020.07.018
Ada Chan, Shaun M. Fallat, S. Kirkland, J. Lin, S. Nasserasr, S. Plosker
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引用次数: 1
A combinatorial identity for the p-binomialcoefficient based on abelian groups 基于阿贝尔群的p-二项式效率的组合恒等式
Pub Date : 2019-12-23 DOI: 10.2140/MOSCOW.2021.10.13
C. Kumar
For a non-negative integer $k$ and a positive integer $n$ with $kleq n$, we prove a combinatorial identity for the $p$-binomial coefficient $binom{n}{k}_p$ based on abelian groups. A purely combinatorial proof is not known for this identity. While proving this identity, for $r,sin mathbb{N}$ and $p$ a prime, we present a purely combinatorial formula for the number of subgroups of $mathbb{Z}^s$ of finite index $p^r$ with quotient isomorphic to the finite abelian $p$-group of type $underline{lambda}$ a partition of $r$ into at most $s$ parts. This purely combinatorial formula is similar to the combinatorial formula for subgroups of a certain type in a finite abelian $p$-group obtained by Lynne Marie Butler. As consequences, this combinatorial formula gives rise many enumeration formulae which are polynomial in $p$ with non-negative integer coefficients.
对于一个非负整数$k$和一个带$kleq n$的正整数$n$,我们证明了基于阿贝尔群的$p$ -二项式系数$binom{n}{k}_p$的一个组合恒等式。对于这个恒等式,没有一个纯粹的组合证明。在证明这个恒等式的同时,对于$r,sin mathbb{N}$和$p$ a素数,我们给出了有限索引$p^r$的$mathbb{Z}^s$的子群数目的一个纯组合公式,这些子群商同构于$r$的$underline{lambda}$ a划分为最多$s$个部分的有限阿贝尔$p$ -群。这个纯组合公式类似于Lynne Marie Butler得到的有限阿贝尔$p$ -群中某类型子群的组合公式。因此,这个组合公式产生了许多在$p$中多项式的非负整数系数的枚举公式。
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引用次数: 0
Techniques for determining equality of the maximum nullity and the zero forcing number of a graph 确定图的最大零值和零强迫数相等的技术
Pub Date : 2019-12-16 DOI: 10.13001/ELA.2021.4967
Derek Young
It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph. In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters which bound the maximum nullity of a graph below. In particular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdi'ere type parameter and the vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.
已知图的零强迫数是图的最大零值的上界。本文通过研究限定图的最大零值的各种图参数,寻找保证图的最大零值与图的强制零值相同的特征。特别地,我们引入了一个新的图参数作为图的最大零值的下界。结果表明,阿兹特克菱形图的最大零值和零强迫数是相同的。考虑的其他图形参数包括Colin de Verdi'ere类型参数和顶点连通性。我们还使用矩阵,如图的除数矩阵和图的邻接矩阵的公平划分,来建立图的邻接矩阵的零的下界。
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引用次数: 0
Maximally nonassociative quasigroups via quadratic orthomorphisms 通过二次正构的最大非结合拟群
Pub Date : 2019-12-15 DOI: 10.5802/alco.165
A. Drápal, Ian M. Wanless
A quasigroup $Q$ is said to be emph{maximally nonassociative} if $xcdot (ycdot z) = (xcdot y)cdot z$ implies $x=y=z$, for all $x,y,zin Q$. We show that, with finitely many exceptions, there exists a maximally nonassociative quasigroup of order $n$ whenever $n$ is not of the form $n=2p_1$ or $n=2p_1p_2$ for primes $p_1,p_2$ with $p_1le p_2<2p_1$.
对于所有的$x,y,zin Q$,如果$xcdot (ycdot z) = (xcdot y)cdot z$暗示$x=y=z$,则称一个拟群$Q$是emph{最大非结合}的。我们证明,除了有限的例外情况,对于含有$p_1le p_2<2p_1$的质数$p_1,p_2$,只要$n$不是$n=2p_1$或$n=2p_1p_2$的形式,就存在一个阶为$n$的最大非结合拟群。
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引用次数: 5
Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity 对称Grothendieck多项式的度与Castelnuovo-Mumford正则性
Pub Date : 2019-12-10 DOI: 10.1090/proc/15294
Jenna Rajchgot, Yifei Ren, Colleen Robichaux, Avery St. Dizier, Anna Weigandt
We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Grassmannian Schubert varieties.
我们给出了Grassmannian置换的Grothendieck多项式的阶的一个显式公式和Grassmannian置换的Schubert行列式理想的Castelnuovo-Mumford正则的一个密切相关的公式。然后,我们提供了Kummini-Lakshmibai-Sastry-Seshadri关于特定Grassmannian Schubert变的标准开块的规律性公式的猜想的一个反例,并表明我们的工作在这些情况下产生了一个替代的显式公式。最后给出了关于任意格拉斯曼-舒伯特变的标准开块的规律性的一个新猜想。
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引用次数: 13
期刊
arXiv: Combinatorics
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