Discrete Mathematics最新文献

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Distance-regular graphs with classical parameters that support a uniform structure: Case q ≥ 2 具有支持统一结构的经典参数的距离规则图：情况 q ≥ 2

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-18 DOI: 10.1016/j.disc.2024.114263

Let $Γ = (X, R)$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set $R$ . Fix a vertex $x \in X$ , and define $R_{f} = R ∖ {y z | \partial (x, y) = \partial (x, z)}$ , where ∂ denotes the path-length distance in Γ. Observe that the graph $Γ_{f} = (X, R_{f})$ is bipartite. We say that Γ supports a uniform structure with respect to x whenever $Γ_{f}$ has a uniform structure with respect to x in the sense of Miklavič and Terwilliger [7].

Assume that Γ is a distance-regular graph with classical parameters $(D, q, α, β)$ and diameter $D \geq 4$ . Recall that q is an integer such that $q \notin {- 1, 0}$ . The purpose of this paper is to study when Γ supports a uniform structure with respect to x. We studied the case $q \leq 1$ in [3], and so in this paper we assume $q \geq 2$ . Let $T = T (x)$ denote the Terwilliger algebra of Γ with respect to x. Under an additional assumption that every irreducible T-module with endpoint 1 is thin, we show that if Γ supports a uniform structure with respect to x, then either $α = 0$ or $α = q$ , $β = q^{2} (q^{D} - 1) / (q - 1)$ , and $D \equiv 0 (\mod 6)$ .

让Γ=(X,R) 表示具有顶点集 X 和边集 R 的有限、简单、连通和不定向的非双向图。固定一个顶点 x∈X，定义 Rf=R∖{yz|∂(x,y)=∂(x,z)}，其中∂表示Γ中的路径长度距离。请注意，图 Γf=(X,Rf) 是双向的。假设 Γ 是一个距离规则图，其经典参数为 (D,q,α,β)，直径为 D≥4。回顾一下，q 是一个整数，使得 q∉{-1,0}。我们在 [3] 中研究过 q≤1 的情况，因此本文假设 q≥2 。让 T=T(x) 表示 Γ 关于 x 的泰尔维利格代数。在每个端点为 1 的不可还原 T 模块都是薄的这一额外假设下，我们证明了如果 Γ 支持关于 x 的均匀结构，那么要么 α=0 要么 α=q，β=q2(qD-1)/(q-1)，D≡0(mod6)。

引用次数: 0

New results on asymmetric orthogonal arrays with strength t ≥ 3 强度 t ≥ 3 的非对称正交阵列的新结果

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-16 DOI: 10.1016/j.disc.2024.114264

The orthogonal array holds significant importance as a research topic within the realms of combinatorial design theory and experimental design theory, with widespread applications in statistics, computer science, coding theory and cryptography. This paper presents three constructions for asymmetric orthogonal arrays including juxtaposition, generator matrices over Galois fields and mixed difference matrices. Subsequently, many new infinite families of asymmetric orthogonal arrays with strength $t \geq 3$ are obtained. Furthermore, some new infinite families of large sets of orthogonal arrays with mixed levels are also obtained.

正交阵列作为组合设计理论和实验设计理论领域的一个重要研究课题，在统计学、计算机科学、编码理论和密码学中有着广泛的应用。本文介绍了非对称正交阵列的三种构造，包括并列、伽罗瓦域上的生成矩阵和混合差分矩阵。随后，得到了强度 t≥3 的许多新的非对称正交阵列无穷族。此外，还得到了一些新的具有混合水平的大集正交阵列无穷族。

引用次数: 0

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

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

<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

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

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

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

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。

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

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

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

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

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