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arXiv: Combinatorics最新文献

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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
arXiv: Combinatorics Pub Date : 2020-02-20 DOI: 10.22108/TOC.2020.121737.1709
A. Ghalavand, A. Ashrafi
{"title":"On a Conjecture about Degree Deviation Measure of Graphs","authors":"A. Ghalavand, A. Ashrafi","doi":"10.22108/TOC.2020.121737.1709","DOIUrl":"https://doi.org/10.22108/TOC.2020.121737.1709","url":null,"abstract":"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.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88402348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Separating Bohr denseness from measurable recurrence 从可测递归中分离玻尔密度
arXiv: Combinatorics Pub Date : 2020-02-17 DOI: 10.19086/da.26859
John T. Griesmer
{"title":"Separating Bohr denseness from measurable recurrence","authors":"John T. Griesmer","doi":"10.19086/da.26859","DOIUrl":"https://doi.org/10.19086/da.26859","url":null,"abstract":"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.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86564939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Demazure crystals for Kohnert polynomials Kohnert多项式的形变晶体
arXiv: Combinatorics Pub Date : 2020-02-17 DOI: 10.1090/tran/8560
Sami H. Assaf
{"title":"Demazure crystals for Kohnert polynomials","authors":"Sami H. Assaf","doi":"10.1090/tran/8560","DOIUrl":"https://doi.org/10.1090/tran/8560","url":null,"abstract":"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.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77624983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv: Combinatorics Pub Date : 2020-02-14 DOI: 10.5802/alco.113
J. McKee, C. Smyth
{"title":"Symmetrizable integer matrices having all their eigenvalues in the interval $[-2,2]$","authors":"J. McKee, C. Smyth","doi":"10.5802/alco.113","DOIUrl":"https://doi.org/10.5802/alco.113","url":null,"abstract":"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. \u0000In 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).","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89920608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Arithmetic combinatorics on Vinogradov systems 维诺格拉多夫系统的算术组合
arXiv: Combinatorics Pub Date : 2020-01-22 DOI: 10.1090/tran/8121
Akshat Mudgal
{"title":"Arithmetic combinatorics on Vinogradov systems","authors":"Akshat Mudgal","doi":"10.1090/tran/8121","DOIUrl":"https://doi.org/10.1090/tran/8121","url":null,"abstract":"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.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75399974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Complex Hadamard diagonalisable graphs 复哈达玛可对角图
arXiv: Combinatorics 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
{"title":"Complex Hadamard diagonalisable graphs","authors":"Ada Chan, Shaun M. Fallat, S. Kirkland, J. Lin, S. Nasserasr, S. Plosker","doi":"10.1016/j.laa.2020.07.018","DOIUrl":"https://doi.org/10.1016/j.laa.2020.07.018","url":null,"abstract":"","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73204337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv: Combinatorics Pub Date : 2019-12-23 DOI: 10.2140/MOSCOW.2021.10.13
C. Kumar
{"title":"A combinatorial identity for the p-binomial\u0000coefficient based on abelian groups","authors":"C. Kumar","doi":"10.2140/MOSCOW.2021.10.13","DOIUrl":"https://doi.org/10.2140/MOSCOW.2021.10.13","url":null,"abstract":"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.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78338612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv: Combinatorics Pub Date : 2019-12-16 DOI: 10.13001/ELA.2021.4967
Derek Young
{"title":"Techniques for determining equality of the maximum nullity and the zero forcing number of a graph","authors":"Derek Young","doi":"10.13001/ELA.2021.4967","DOIUrl":"https://doi.org/10.13001/ELA.2021.4967","url":null,"abstract":"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.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87257531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv: Combinatorics Pub Date : 2019-12-15 DOI: 10.5802/alco.165
A. Drápal, Ian M. Wanless
{"title":"Maximally nonassociative quasigroups via quadratic orthomorphisms","authors":"A. Drápal, Ian M. Wanless","doi":"10.5802/alco.165","DOIUrl":"https://doi.org/10.5802/alco.165","url":null,"abstract":"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 $p_1le p_2<2p_1$.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74371172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
arXiv: Combinatorics Pub Date : 2019-12-10 DOI: 10.1090/proc/15294
Jenna Rajchgot, Yifei Ren, Colleen Robichaux, Avery St. Dizier, Anna Weigandt
{"title":"Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity","authors":"Jenna Rajchgot, Yifei Ren, Colleen Robichaux, Avery St. Dizier, Anna Weigandt","doi":"10.1090/proc/15294","DOIUrl":"https://doi.org/10.1090/proc/15294","url":null,"abstract":"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.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75414802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
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