Pub Date : 2020-02-20DOI: 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。
{"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":"49 1","pages":""},"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}
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.
{"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":"35 1","pages":""},"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}
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.
{"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":"105 1","pages":""},"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}
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).
{"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":"13 1","pages":""},"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}
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.
{"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":"17 1","pages":""},"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}
Pub Date : 2020-01-01DOI: 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":"74 1","pages":""},"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}
Pub Date : 2019-12-23DOI: 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。
{"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":"10 1","pages":""},"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}
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。
{"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":"20 1","pages":""},"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}
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 $p_1le p_2<2p_1$.
{"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":"282 1","pages":""},"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}
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.
{"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":"16 1","pages":""},"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}
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