Discrete Mathematics最新文献

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The weight hierarchies of linear codes from simplicial complexes 来自简单复合物的线性编码的权重层次

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-12 DOI: 10.1016/j.disc.2024.114240

Chao Liu, Dabin Zheng , Wei Lu, Xiaoqiang Wang

The study of the generalized Hamming weight of linear codes is a significant research topic in coding theory as it conveys the structural information of the codes and determines their performance in various applications. However, determining the generalized Hamming weights of linear codes, especially the weight hierarchy, is generally challenging. In this paper, we investigate the generalized Hamming weights of a class of linear code $C$ over $F_{q}$ , which is constructed from defining sets. These defining sets are either special simplicial complexes or their complements in $F_{q}^{m}$ . We determine the complete weight hierarchies of these codes by analyzing the maximum or minimum intersection of certain simplicial complexes and all r-dimensional subspaces of $F_{q}^{m}$ , where $1 \leq r \leq \dim_{F_{q}} (C)$ .

线性编码的广义汉明权重传递了编码的结构信息，决定了编码在各种应用中的性能，因此研究线性编码的广义汉明权重是编码理论中的一个重要研究课题。然而，确定线性编码的广义汉明权重，尤其是权重层次结构，通常具有挑战性。本文研究了一类 Fq 上线性编码 C 的广义汉明权重，该编码由定义集构建。这些定义集要么是特殊的单纯复数，要么是它们在 Fqm 中的补集。我们通过分析某些单纯复数与 Fqm 的所有 r 维子空间的最大或最小交集（其中 1≤r≤dimFq(C) ），确定这些代码的完整权重等级。

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引用次数: 0

On the restricted order of asymptotic bases 关于渐近基的限制阶

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-12 DOI: 10.1016/j.disc.2024.114260

Jin-Hui Fang, Ying Cheng

<div>Let <math><mi>N</mi></math> be the set of all positive integers. For a set A of positive integers, let <math><mi>A</mi><mo>∼</mo><mi>N</mi></math> denote that A contains all but finitely many positive integers. For an integer <math><mi>h</mi><mo>⩾</mo><mn>2</mn></math>, define <math><mi>h</mi><mi>A</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>:</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>∈</mo><mi>A</mi><mo>}</mo></math> and <math><mi>h</mi><mo>×</mo><mi>A</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>:</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>∈</mo><mi>A</mi><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≠</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub></math> for <math><mi>i</mi><mo>≠</mo><mi>j</mi><mo>}</mo></math>. In 2023, Chen and Yu [Discrete Math. 346 (2023), Paper No. 113388.] proved that, there exists a set B of positive integers such that: <math><msub><mrow><mi>lim</mi></mrow><mrow><mi>x</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo>⁡</mo><mi>B</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>/</mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math>, <math><mi>B</mi><mo>⋃</mo><mo>(</mo><mn>2</mn><mi>B</mi><mo>)</mo><mo>∼</mo><mi>N</mi></math>, <math><mi>B</mi><mo>⋃</mo><mo>(</mo><mn>2</mn><mo>×</mo><mi>B</mi><mo>)</mo><mo>≁</mo><mi>N</mi></math>, and <math><mi>B</mi><mo>⋃</mo><mo>(</mo><mn>2</mn><mo>×</mo><mi>B</mi><mo>)</mo><mo>⋃</mo><mo>(</mo><mn>3</mn><mo>×</mo><mi>B</mi><mo>)</mo><mo>∼</mo><mi>N</mi></math>. In this paper, we construct a somewhat dense set B satisfying the above properties. That is, there exists a set B of positive integers such that: <math><msub><mrow><mrow><mi>lim</mi></mrow><mspace></mspace><mrow><mi>inf</mi></mrow></mrow><mrow><mi>x</mi><mo>→</mo><mo>∞</mo></mrow></msub><mspace></mspace><mi>B</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>/</mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math>, <math><msub><mrow><mrow><mi>lim</mi></mrow><mspace></mspace><mrow><mi>sup</mi></mrow></mrow><mrow><mi>x</mi><mo>→</mo><mo>∞</mo></mrow></msub><mspace></mspace><mi>B</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>/</mo><mi>x</mi><mo>=</mo><mn>1</mn></math>, <math><mi>B</mi><mo>⋃</mo><mo>(</mo><mn>2</mn><mi>B</mi><mo>)</mo><mo>∼</mo><mi>N</mi></math>, <math><mi>B</mi><mo>⋃</mo><mo>(</mo><m

设 N 是所有正整数的集合。对于一个正整数集合 A，让 A∼N 表示 A 包含所有但不超过有限个的正整数。对于整数 h⩾2，定义 hA={a1+⋯+ah:a1,⋯,ah∈A} 和 h×A={a1+⋯+ah:a1,⋯,ah∈A,ai≠aj for i≠j} 。2023 年，Chen 和 Yu [Discrete Math. 346 (2023)，Paper No. 113388.] 证明，存在一个正整数集合 B，使得：limx→∞B(x)/x=1/2，B⋃(2B)∼N，B⋃(2×B)≁N，且 B⋃(2×B)⋃(3×B)∼N。在本文中，我们将构造一个满足上述性质的略密集 B。也就是说，存在一个正整数集合 B，使得：liminfx→∞B(x)/x=1/2，limsupx→∞B(x)/x=1，B⋃(2B)∼N，B⋃(2×B)≁N，且 B⋃(2×B)⋃(3×B)∼N。

引用次数: 0

Corrigendum to “Singular graphs and the reciprocal eigenvalue property” [Discrete Math. 347 (2024) 114003] 奇异图和倒易特征值性质 "的更正[离散数学. 347 (2024) 114003]

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-12 DOI: 10.1016/j.disc.2024.114198

Sasmita Barik , Debabrota Mondal , Sukanta Pati , Kuldeep Sarma

We correct an error in Theorem 13 in our published article Barik et al. [1].

我们纠正了已发表文章 Barik 等人 [1] 中定理 13 中的一个错误。

引用次数: 0

Oriented posets, rank matrices and q-deformed Markov numbers 定向正集、秩矩阵和 q 变形马尔可夫数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-12 DOI: 10.1016/j.disc.2024.114256

Ezgi Kantarcı Oğuz

We define oriented posets with corresponding rank matrices, where linking two posets by an edge corresponds to matrix multiplication. In particular, linking chains via this method gives us fence posets, and taking traces gives us circular fence posets. As an application, we give a combinatorial model for q-deformed Markov numbers. We also resolve a conjecture of Leclere and Morier-Genoud and give several identities between circular rank polynomials.

我们定义了具有相应秩矩阵的定向集合，通过边将两个集合连接起来就相当于矩阵乘法。特别是，通过这种方法连接链可以得到栅栏集合，而通过迹则可以得到循环栅栏集合。作为应用，我们给出了 q 变形马尔可夫数的组合模型。我们还解决了勒克莱尔和莫里埃-杰努德的一个猜想，并给出了循环秩多项式之间的几个同分异构体。

引用次数: 0

Diversity and intersecting theorems for weak compositions 弱组合的多样性和相交定理

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-11 DOI: 10.1016/j.disc.2024.114250

Cheng Yeaw Ku , Kok Bin Wong

Let $N_{0}$ be the set of non-negative integers, and let $P (n, k)$ denote the set of all weak compositions of n with k parts, i.e., $P (n, k) = {(x_{1}, x_{2}, \dots, x_{k}) \in N_{0}^{k} : x_{1} + x_{2} + \dots + x_{k} = n}$ . For any element $u = (u_{1}, u_{2}, \dots, u_{k}) \in P (n, k)$ , denote its ith-coordinate by $u (i)$ , i.e., $u (i) = u_{i}$ . A family $A \subseteq P (n, k)$ is said to be t-intersecting if $| {i : u (i) = v (i)} | \geq t$ for all $u, v \in A$ . In this paper, we consider the diversity and other intersecting theorems for weak compositions.

设 N0 为非负整数集合，P(n,k) 表示 n 的所有 k 部分的弱合成集合，即 P(n,k)={(x1,x2,...,xk)∈N0k:x1+x2+⋯+xk=n}。对于任何元素 u=（u1,u2,...,uk）∈P（n,k），用 u(i) 表示其 ith 坐标，即 u(i)=ui 。对于所有 u，v∈A，如果|{i:u(i)=v(i)}|≥t，则称一个族 A⊆P(n,k)为 t 交族。在本文中，我们将考虑弱组合的多样性和其他相交定理。

引用次数: 0

Bounding the intersection number c2 of a distance-regular graph with classical parameters (D,b,α,β) in terms of b 用 b 限定具有经典参数 (D,b,α,β)的距离规则图的交点数 c2

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-11 DOI: 10.1016/j.disc.2024.114239

Jack H. Koolen , Chenhui Lv , Jongyook Park , Qianqian Yang

Let Γ be a distance-regular graph with classical parameters $(D, b, α, β)$ and $b \geq 1$ . It is known that Γ is Q-polynomial with respect to $θ_{1}$ , where $θ_{1} = \frac{b_{1}}{b} - 1$ is the second largest eigenvalue of Γ. And it was shown that for a distance-regular graph Γ with classical parameters $(D, b, α, β)$ , $D \geq 5$ and $b \geq 1$ , if $a_{1}$ is large enough compared to b and Γ is thin, then the intersection number $c_{2}$ of Γ is bounded above by a function of b. In this paper, we obtain a similar result without the assumption that the graph Γ is thin.

设 Γ 是一个距离规则图，其经典参数为 (D,b,α,β)，且 b≥1 已知 Γ 关于 θ1 是 Q 多项式，其中 θ1=b1b-1 是 Γ 的第二大特征值。本文在不假设图 Γ 很薄的情况下也得到了类似的结果。

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引用次数: 0

A digraph version of the Friendship Theorem 友谊定理的数字图版本

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-11 DOI: 10.1016/j.disc.2024.114238

Myungho Choi, Hojin Chu, Suh-Ryung Kim

The Friendship Theorem states that if in a party any pair of persons has precisely one common friend, then there is always a person who is everybody's friend and the theorem has been proved by Paul Erdős, Alfréd Rényi, and Vera T. Sós in 1966. “What would happen if instead any pair of persons likes precisely one person?” While a friendship relation is symmetric, a liking relation may not be symmetric. Therefore to represent a liking relation we should use a directed graph. We call this digraph a “liking digraph”. It is easy to check that a symmetric liking digraph becomes a friendship graph if each directed cycle of length two is replaced with an edge. In this paper, we provide a digraph formulation of the Friendship Theorem which characterizes the liking digraphs. We also establish a sufficient and necessary condition for the existence of liking digraphs.

友谊定理指出，如果在一个聚会中，任何一对人恰好有一个共同的朋友，那么总有一个人是大家的朋友，该定理由保罗-厄多斯、阿尔弗雷德-雷尼和维拉-索斯于 1966 年证明。"如果任何一对人恰好喜欢一个人，会发生什么呢？"友谊关系是对称的，而喜欢关系可能不是对称的。因此，我们应该使用有向图来表示喜欢关系。我们称这种有向图为 "喜欢有向图"。如果把每个长度为 2 的有向循环替换为一条边，就可以很容易地检验出对称的喜好数图变成了友谊图。在本文中，我们提供了 "友谊定理 "的数图表述，它描述了喜欢数图的特征。我们还建立了存在喜欢数图的充分必要条件。

引用次数: 0

Extremal graphs for the odd prism 奇数棱柱的极值图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-11 DOI: 10.1016/j.disc.2024.114249

Xiaocong He , Yongtao Li , Lihua Feng

The Turán number $ex (n, H)$ of a graph H is the maximum number of edges in an n-vertex graph which does not contain H as a subgraph. The Turán number of regular polyhedrons was widely studied in a series of works due to Simonovits. In this paper, we shall present the exact Turán number of the prism $C_{2 k + 1}^{□}$ , which is defined as the Cartesian product of an odd cycle $C_{2 k + 1}$ and an edge $K_{2}$ . Applying a deep theorem of Simonovits and a stability result of Yuan (2022) [55], we shall determine the exact value of $ex (n, C_{2 k + 1}^{□})$ for every $k \geq 1$ and sufficiently large n, and we also characterize the extremal graphs. Moreover, in the case of $k = 1$ , motivated by a recent result of Xiao et al. (2022) [49], we will determine the exact value of $ex (n, C_{3}^{□})$ for every n instead of for sufficiently large n.

图 H 的图兰数 ex(n,H) 是 n 个顶点图中不包含 H 作为子图的最大边数。西蒙诺维茨在一系列著作中对正多面体的图兰数进行了广泛研究。在本文中，我们将提出棱 C2k+1□ 的精确图兰数，它被定义为奇数循环 C2k+1 与边 K2 的笛卡尔积。应用 Simonovits 的深层定理和 Yuan (2022) 的稳定性结果[55]，我们将确定每 k≥1 且 n 足够大时 ex(n,C2k+1□) 的精确值，并描述极值图的特征。此外，在 k=1 的情况下，受肖等人（2022）的最新结果[49]的启发，我们将确定每个 n 而不是足够大的 n 的 ex(n,C3□) 的精确值。

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引用次数: 0

The Jacobian of a graph and graph automorphisms 图的雅可比和图的自动变形

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-11 DOI: 10.1016/j.disc.2024.114259

István Estélyi , Ján Karabáš , Alexander Mednykh , Roman Nedela

In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph X in the group of symmetries of the Jacobian of X. As a consequence we show that if a 3-edge-connected graph X admits a nonabelian semiregular group of automorphisms, then the Jacobian of X cannot be cyclic. In particular, Cayley graphs of degree at least three arising from nonabelian groups have non-cyclic Jacobians. While the size of the Jacobian of X is well-understood – it is equal to the number of spanning trees of X – the combinatorial interpretation of the rank of Jacobian of a graph is unknown. Our paper presents a contribution in this direction.

在本文中，我们研究了图 X 的自变量群的某些线性表示在 X 的雅各布对称群中的忠实性。结果表明，如果一个三边连接的图 X 接受一个非阿贝尔半圆自变量群，那么 X 的雅各布不可能是循环的。特别是，由非阿贝尔群产生的阶数至少为 3 的 Cayley 图具有非循环雅各布。虽然 X 的雅各布的大小很好理解--它等于 X 的生成树的数量，但对图的雅各布秩的组合解释还不清楚。我们的论文在这方面做出了贡献。

引用次数: 0

Adversarial graph burning densities 逆向图燃烧密度

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-11 DOI: 10.1016/j.disc.2024.114253

Karen Gunderson , William Kellough , J.D. Nir , Hritik Punj

Graph burning is a discrete-time process that models the spread of influence in a network. Vertices are either burning or unburned, and in each round, a burning vertex causes all of its neighbours to become burning before a new fire source is chosen to become burning. We introduce a variation of this process that incorporates an adversarial game played on a nested, growing sequence of graphs. Two players, Arsonist and Builder, play in turns: Builder adds a certain number of new unburned vertices and edges incident to these to create a larger graph, then every vertex neighbouring a burning vertex becomes burning, and finally Arsonist ‘burns’ a new fire source. This process repeats forever. Arsonist is said to win if the limiting fraction of burning vertices tends to 1, while Builder is said to win if this fraction is bounded away from 1.

The central question of this paper is determining if, given that Builder adds $f (n)$ vertices at turn n, either Arsonist or Builder has a winning strategy. In the case that $f (n)$ is asymptotically polynomial, we give threshold results for which player has a winning strategy.

图燃烧是一个离散时间过程，是网络中影响力传播的模型。顶点要么燃烧，要么未燃烧，在每一轮中，一个燃烧的顶点会导致其所有相邻顶点燃烧，然后再选择一个新的火源燃烧。我们引入了这一过程的变体，在嵌套的、不断增长的图序列中加入了对抗游戏。两名玩家（纵火者和建造者）轮流进行游戏：建造者添加一定数量的新的未燃烧顶点和与之相连的边，以创建一个更大的图，然后每个与燃烧顶点相邻的顶点都变成燃烧顶点，最后纵火犯 "燃烧 "一个新的火源。这个过程永远重复。如果燃烧顶点的极限分数趋向于 1，那么 "纵火者 "就获胜了；如果这个分数远离 1，那么 "建造者 "就获胜了。本文的核心问题是确定，如果 "建造者 "在第 n 轮增加了 f(n) 个顶点，那么 "纵火者 "或 "建造者 "是否都有获胜策略。在 f(n) 是渐近多项式的情况下，我们给出了哪一方有获胜策略的阈值结果。

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