Discrete Mathematics最新文献

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Turán number of complete bipartite graphs with bounded matching number Turán匹配数有界的完全二部图数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-04-25 DOI: 10.1016/j.disc.2025.114552

Huan Luo, Xiamiao Zhao, Mei Lu

<div><div>Let <math><mi>F</mi></math> be a family of graphs. A graph G is <math><mi>F</mi></math>-free if G does not contain any <math><mi>F</mi><mo>∈</mo><mi>F</mi></math> as a subgraph. The Turán number <math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> is the maximum number of edges in an n-vertex <math><mi>F</mi></math>-free graph. Let <math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub></math> be the matching consisting of s independent edges. Recently, Alon and Frankl determined the exact value of <math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math>. Gerbner obtained several results about <math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><mi>F</mi><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math> when F satisfies certain properties. In this paper, we determine the exact value of <math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math> when <math><mi>s</mi><mo>,</mo><mi>n</mi></math> are large enough for every <math><mn>3</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>t</mi></math>. When n is large enough, we also show that <math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo><mo>=</mo><mi>n</mi><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>s</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></math> for <math><mi>s</mi><mo>≥</mo><mn>12</mn></math> and <math><mtext>ex</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>t</mi></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo><mo>=</mo><mi>n</mi><mo>+</mo><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>s</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></math> when <math><mi>t</mi><mo>≥</mo><mn>3</mn></math> and s is large enough.</div><

设F是一个图族。如果图G不包含任何F∈F作为子图，则图G是F自由的。Turán数字ex（n，F）是一个有n顶点的无F图的最大边数。设Ms为由s条独立边组成的匹配。最近，Alon和Frankl确定了ex（n,{Km,Ms+1}）的确切值。Gerbner得到了当F满足某些性质时ex（n,{F,Ms+1}）的几个结果。在本文中，我们确定了当s，n足够大且每3≤r≤t时ex（n,{K,t, m +1}）的精确值。当n足够大时，我们还证明了当s≥12时ex(n,{K2,2,Ms+1})=n+(s2)−≤≤2s；当t≥3且s足够大时，ex(n,{K2,t,Ms+1})=n+(t−1)(s2)−≤≤2s；

引用次数: 0

On the roots of maximal matching polynomials 关于极大匹配多项式的根

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-04-24 DOI: 10.1016/j.disc.2025.114547

Matt Burnham, Aysel Erey

A maximal matching of a graph G is a matching of G which is not properly contained in any other matching of G. Let

m_{k} (G)

be the number of maximal matchings of size k in G. The maximal matching polynomial of G is defined by

m (G; x) = \sum_{k} m_{k} (G) x^{k}

. It is known that maximal matching polynomials generalize the well-known matching polynomials, as the matching polynomial of every graph can be obtained from the maximal matching polynomial of some other graph. While the roots of matching polynomials have been extensively studied and well understood, the study of the roots of maximal matching polynomials has not been developed. In this article, we study the location of the roots of these polynomials. We show that maximal matching polynomials of paths and cycles have only real roots, and provide interlacing relations for their roots. On the other hand, unlike matching polynomials, maximal matching polynomials can have non-real roots, and we provide an infinite family of graphs whose maximal matching polynomials have non-real roots.

让 mk(G) 表示 G 中大小为 k 的最大匹配数。G 的最大匹配多项式定义为 m(G;x)=∑kmk(G)xk。众所周知，最大匹配多项式概括了众所周知的匹配多项式，因为每个图的匹配多项式都可以从另一些图的最大匹配多项式得到。虽然匹配多项式的根已经得到了广泛的研究和很好的理解，但对最大匹配多项式的根的研究还没有发展起来。在本文中，我们将研究这些多项式根的位置。我们证明了路径和循环的最大匹配多项式只有实数根，并为它们的根提供了交错关系。另一方面，与匹配多项式不同，最大匹配多项式可以有非实根，我们提供了最大匹配多项式有非实根的无限图族。

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引用次数: 0

Descents in powers of permutations 排列幂的下降

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-04-24 DOI: 10.1016/j.disc.2025.114551

Kassie Archer, Aaron Geary

We consider a few special cases of the more general question: How many permutations

π \in S_{n}

have the property that

π^{2}

has j descents for some j? In this paper, we first enumerate Grassmannian permutations π by the number of descents in

π^{2}

. We then consider all permutations whose square has exactly one descent, fully enumerating when the descent is “small” and providing a lower bound in the general case. Finally, we enumerate permutations whose square or cube has the maximum number of descents, and finish the paper with a few future directions for study.

我们考虑一些更一般的问题的特殊情况：π∈Sn有多少个排列π∈Sn有j的下降？在本文中，我们首先用π2的下降数来枚举格拉斯曼排列π。然后，我们考虑所有平方只有一次下降的排列，充分列举下降“小”的情况，并提供一般情况下的下界。最后，我们列举了其平方或立方下降次数最多的排列，并对未来的研究方向进行了展望。

引用次数: 0

Intersecting families with covering number five 与第5号掩体相交的家族

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-04-24 DOI: 10.1016/j.disc.2025.114546

Peter Frankl , Jian Wang

A family

F \subset (\begin{matrix} [n] \\ k \end{matrix})

is called intersecting if any two members of it have non-empty intersection. The covering number of

F

is defined as the minimum integer p such that there exists

T \subset {1, 2, \dots, n}

satisfying

| T | = p

and

T \cap F \neq \emptyset

for all

F \in F

. Define

m (n, k, p)

as the maximum size of an intersecting family

F \subset (\begin{matrix} [n] \\ k \end{matrix})

with covering number at least p. The value of

m (n, k, p)

is only known for

p = 1, 2, 3, 4

. About thirty years ago,

m (n, k, 5)

was determined asymptotically by the first author, Ota and Tokushige. In the present paper, we determine

m (n, k, 5)

for

k \geq 69

and

n \geq 5 k^{6}

如果族F∧([n]k)中的任意两个成员有非空相交，则称其为相交族。F的覆盖数定义为对所有F∈F存在T∧{1,2，…，n}满足|T|=p且T∩F≠∅的最小整数p。定义m（n,k,p）为覆盖数至少为p的相交族F （[n]k）的最大值。m（n,k,p）的值仅在p=1,2,3,4时已知。大约三十年前，m（n,k,5）由第一作者Ota和Tokushige渐近确定。在本文中，我们确定了k≥69和n≥5k6时m（n,k,5）。

引用次数: 0

Group action approaches in Erdős quotient set problem Erdős商集问题中的群作用方法

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-04-24 DOI: 10.1016/j.disc.2025.114505

Will Burstein

<div><div>Let <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> denote the finite field of q elements. Let <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math> be the d-dimensional vector space over the field <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>. Let <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>:</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math>. Let <math><msup><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>:</mo><mo>=</mo><mo>{</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>:</mo><mi>t</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>}</mo></math>. For <math><mi>E</mi><mo>⊂</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math>, denote the distance set by <math><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>{</mo><mo>‖</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>‖</mo><mo>:</mo><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msup><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>−</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>:</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>E</mi><mo>}</mo></math>. Denote the Erdős quotient set by <math><mfrac><mrow><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mrow><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mfrac><mo>:</mo><mo>=</mo><mo>{</mo><mfrac><mrow><mi>s</mi></mrow><mrow><mi>t</mi></mrow></mfrac><mo>:</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo><mo>,</mo><mi>t</mi><mo>≠</mo><mn>0</mn><mo>}</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>.</div><div>The Erdős quotient set problem was introduced in [13] where it was shown that for even <math><mi>d</mi><mo>≥</mo><mn>2</mn></math>, if <math><mi>E</mi><mo>⊂</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math> such that <math><mo>|</mo><mi>E</mi><mo>|</mo><mo>≫</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup></math>, then <math><mfrac><mrow><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mrow><mi>Δ</mi><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mfrac><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>

设Fq表示q个元素的有限域。设Fqd是场Fq上的d维向量空间。让Fq⁎:= Fq∖{0}。让(Fq) 2: = {t2: t∈Fq}。对E⊂Fqd,表示设定的距离Δ(E): ={为x−y为:= (x1−y1) 2 +⋯+ (xd−码)2:x, y∈E}。表示设定的Erdő年代商Δ(E)Δ(E): ={圣:s t∈Δ(E), t≠0}= Fq。在[13]中引入了Erdős商集问题，证明了对于偶d≥2，如果E∧Fq2使得|E|²qd/2，则Δ(E)Δ(E)=Fq。后一个结果的证明是相当复杂的，并且在[17]中，对于q≡3（mod4）和d=2的情况，使用群作用方法得到了一个简单的证明。在问≡3 (mod4)设置,为每个r∈(Fq) 2,[17]显示如果E⊂Fq2, V (r): = # {(a, b, c, d)∈E4:为−b为为c−d为= r}≫| | 4 q。在这项工作中，我们在d=2的q≡3（mod4）设置中使用群作用技术，并通过去除r∈(Fq)2的假设来改进[17]的结果。具体地说，如果d=2, q≡3(mod4)，且|E|≥2q，则对于每个r∈Fq，则V(r)≥|E|42q。最后，我们利用商集结果的证明技术改进了[2]的主要结果。我们新颖地引入了矩阵Aeven和Aodd，这是改进[17]，[2]结果的关键。

引用次数: 0

Bounds for asymmetric nested orthogonal arrays 非对称嵌套正交数组的界

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-04-22 DOI: 10.1016/j.disc.2025.114549

Xiao Lin, Shanqi Pang, Guangzhou Chen

Nested orthogonal arrays (NOAs) are more and more widely used in diverse experiments. An important problem in the study of NOAs is to determine the minimal number of runs, i.e., to find the bounds on the rows for NOAs. These bounds are quite powerful in proving nonexistence. Although the bounds for symmetric NOAs were derived over a decade, the bounds for asymmetric NOAs remain an open problem. This article presents the bounds for asymmetric NOAs.

嵌套正交阵列在各种实验中得到越来越广泛的应用。noa研究中的一个重要问题是确定最小运行次数，即找到noa的行边界。这些界限在证明不存在性方面是相当有力的。尽管对称noa的边界已经推导了十多年，但非对称noa的边界仍然是一个开放的问题。本文介绍了非对称noa的边界。

引用次数: 0

Upper bound for the number of maximal dissociation sets in trees 树中最大解离集数目的上界

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-04-22 DOI: 10.1016/j.disc.2025.114545

Ziyuan Wang , Lei Zhang , Jianhua Tu , Liming Xiong

Let G be a simple graph. A dissociation set of G is defined as a set of vertices that induces a subgraph in which every vertex has a degree of at most 1. A dissociation set is maximal if it is not contained as a proper subset in any other dissociation set. We introduce the notation

Φ (G)

to represent the number of maximal dissociation sets in G. This study focuses on trees, specifically showing that for any tree T of order

n \geq 4

, the following inequality holds:

Φ (T) \leq 3^{\frac{n - 1}{3}} + \frac{n - 1}{3} .

We also identify extremal trees that attain this upper bound. Additionally, to establish the upper bound on the number of maximal dissociation sets in trees of order n, we also determine the second largest number of maximal dissociation sets in forests of order n.

设G是一个简单的图。解离集G被定义为一组顶点，这些顶点诱导出一个子图，其中每个顶点的度数最多为1。如果一个解离集不作为适当子集包含在任何其他解离集中，则该解离集是最大的。我们引入Φ(G)符号来表示G中最大解离集的个数。本研究着重于树，具体表明对于n≥4阶的任意树T，以下不等式成立：Φ(T)≤3n−13+n−13。我们还找出了达到这个上界的极值树。此外，为了确定n阶树中最大解离集数量的上界，我们还确定了n阶森林中第二大最大解离集的数量。

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引用次数: 0

Constructing cospectral graphs via regular rational orthogonal matrix with level two and three 通过二级和三级规则有理正交矩阵构建余谱图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-04-16 DOI: 10.1016/j.disc.2025.114542

Lihuan Mao, Fu Yan

Two graphs G and H are cospectral if their adjacency matrices share the same spectrum. Constructing cospectral non-isomorphic graphs has been studied extensively for many years and various constructions are known in the literature, e.g. the famous GM-switching method. In this paper, we shall construct cospectral graphs via regular rational orthogonal matrix Q with level two and three. We provide two straightforward algorithms to characterize the adjacency matrix A of the graph G such that

Q^{T} A Q

is again a (0,1)-matrix, and introduce two new switching methods to construct families of cospectral graphs which generalized the GM-switching to some extent.

如果两个图G和H的邻接矩阵具有相同的谱，则它们是共谱。构造同谱非同构图已经被广泛研究了很多年，文献中有各种各样的构造方法，如著名的gm开关法。本文利用二阶和三阶正则有理正交矩阵Q构造同谱图。我们提供了两种简单的算法来表征图G的邻接矩阵A，使QTAQ再次成为(0,1)-矩阵，并引入了两种新的交换方法来构造共谱图族，在一定程度上推广了gm -交换。

引用次数: 0

More results on the spectral radius of graphs with no odd wheels 关于无奇轮图谱半径的更多结果

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-04-16 DOI: 10.1016/j.disc.2025.114550

Wenqian Zhang

For a graph G, the spectral radius

λ_{1} (G)

of G is the largest eigenvalue of its adjacency matrix. An odd wheel

W_{2 k + 1}

with

k \geq 2

is a graph obtained from a cycle of order 2k by adding a new vertex connecting to all the vertices of the cycle. Let

SPEX (n, W_{2 k + 1})

be the set of

W_{2 k + 1}

-free graphs of order n with the maximum spectral radius. Very recently, Cioabă, Desai and Tait [4] characterized the graphs in

SPEX (n, W_{2 k + 1})

for sufficiently large n, where

k \geq 2

and

k \neq 4, 5

. And they left the case

k = 4, 5

as a problem. In this paper, we settle this problem. Moreover, we completely characterize the graphs in

SPEX (n, W_{2 k + 1})

when

k \geq 4

is even and

n \equiv 2 (mod 4)

is sufficiently large. Consequently, the graphs in

SPEX (n, W_{2 k + 1})

are characterized completely for any

k \geq 2

and sufficiently large n.

对于图G， G的谱半径λ1(G)是其邻接矩阵的最大特征值。k≥2的奇轮W2k+1是在一个2k阶的循环中添加一个连接到该循环所有顶点的新顶点而得到的图。设SPEX（n,W2k+1）为最大谱半径的n阶无W2k+1图的集合。最近，cioabei， Desai和Tait[4]在足够大的n下，在SPEX（n,W2k+1）中刻画了k≥2且k≠4,5的图。题目把k=4,5的情况留作一个问题。本文解决了这一问题。此外，当k≥4为偶数且n≡2（mod4）足够大时，我们完全刻画了SPEX（n,W2k+1）中的图。因此，当k≥2且n足够大时，SPEX（n,W2k+1）中的图是完全表征的。

引用次数: 0

The c-boomerang uniformity and c-boomerang spectrum of two classes of permutation polynomials over the finite field F2n 有限域F2n上两类置换多项式的c-回旋均匀性和c-回旋谱

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-04-14 DOI: 10.1016/j.disc.2025.114543

Guanghui Li , Xiwang Cao

The boomerang attack developed by Wagner is a cryptanalysis technique against block ciphers. A new theoretical tool, the Boomerang Connectivity Table (BCT) and the corresponding boomerang uniformity were introduced to evaluate the resistance of a block cipher against the boomerang attack. Using a multiplier differential, Stănică (2021) [28] extended the notion of boomerang uniformity to c-boomerang uniformity. In this paper, we focus on two classes of permutation polynomials over

F_{2^{n}}

. For one of these, we show that the c-boomerang uniformity of this function is equal to 1. For the second type of function, we first consider the c-BCT entries. We then explicitly determine the c-boomerang spectrum of this function by means of characters and some techniques in solving equations over

F_{2^{n}}

瓦格纳提出的回旋镖攻击是一种针对块密码的密码分析技术。该研究引入了一种新的理论工具--回旋镖连接表（BCT）和相应的回旋镖均匀性，以评估区块密码对回旋镖攻击的抵抗力。Stănică (2021) [28]利用乘法差分将回旋镖均匀性概念扩展为 c- 回旋镖均匀性。本文重点研究 F2n 上的两类置换多项式。对于其中一类，我们证明了该函数的 c- 回旋处均匀性等于 1。对于第二类函数，我们首先考虑 c-BCT 项。然后，我们通过在 F2n 上求解方程时使用的字符和一些技术，明确确定了该函数的 c-boomerang 频谱。

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引用次数: 0

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