Discrete Applied Mathematics最新文献

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A note on the generalized Turán number of star forests 关于星林广义Turán数目的说明

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-04-15 Epub Date: 2025-12-18 DOI: 10.1016/j.dam.2025.12.032

Yu-Yue Zhang, Jian-Hua Yin

<div><div>The generalized Turán number <math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math> is defined to be the maximum number of copies of a complete graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math> in any <math><mi>H</mi></math>-free graph on <math><mi>n</mi></math> vertices. Let <math><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math> denote the star on <math><mrow><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></math> vertices, and let <math><mrow><mi>k</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></mrow></math> denote the disjoint union of <math><mi>k</mi></math> copies of <math><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math>. Gan et al. and Chase determined <math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math> for all integers <math><mrow><mi>s</mi><mo>≥</mo><mn>3</mn></mrow></math>, <math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math>. Recently, Liu and Yin further investigated the problem of determining the generalized Turán number of star forests. They determined <math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>2</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math> for <math><mrow><mi>s</mi><mo>≥</mo><mn>4</mn></mrow></math>, <math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>3</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math> for <math><mrow><mi>s</mi><mo>≥</mo><mn>5</mn></mrow></math>, <math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math>. Moreover, they also determined <math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>4</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math> for <math><mrow><mi>s</mi><mo>≥</mo><mn>6</mn></mrow></math>, <math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math><mrow><mi>n</mi><mo>≥</mo><mn>1<

广义的Turán数ex（n,Ks，H）被定义为在任意n个顶点上的无H图中完全图k的最大拷贝数。设sz表示在1个顶点上的星，kz表示sz的k个拷贝的不相交并。Gan et al.和Chase确定了所有整数S≥3、r≥1和n≥1的ex（n,Ks，S）。最近，Liu和Yin进一步研究了确定广义Turán星林数的问题。当s≥4、r≥1、n≥1时确定ex(n,Ks,2S)，当s≥5、r≥1、n≥1时确定ex（n,Ks,3S）。此外，他们还确定了s≥6，r≥1和n≥1时的ex（n,Ks,4S）。然而，确定2≤k≤4和3≤s≤k+1的ex（n,Ks, Ks）的问题似乎是困难和具有挑战性的。本文研究了上述问题，并确定了当s=3时ex（n,Ks,2S）和当3≤s≤4时ex（n,Ks,3S）。此外，我们还确定了3≤s≤5时ex（n,Ks,4S）。

引用次数: 0

Forbidden configurations and dominating bicliques in undirected 2-quasi best match graphs 无向2-拟最优匹配图中的禁止构型和支配双链

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-04-15 Epub Date: 2026-01-09 DOI: 10.1016/j.dam.2026.01.001

Annachiara Korchmaros , Peter F. Stadler

2-quasi best match graphs (2qBMGs) are directed graphs that capture a notion of close relatedness in phylogenetics. Here, we investigate the underlying undirected graph of a 2qBMG (un2qBMG) and show that it contains neither a path

P_{l}

nor a cycle

C_{l}

of length

l \geq 6

as an induced subgraph. This property guarantees the existence of specific vertex decompositions with dominating bicliques that provide further insights into their structure.

2-准最佳匹配图（2qbmg）是有向图，它捕获了系统发育中近亲关系的概念。本文研究了一个2qBMG （un2qBMG）的底层无向图，证明了它既不包含路径Pl，也不包含长度为l≥6的循环Cl作为诱导子图。这个性质保证了特定顶点分解的存在，这些顶点分解具有主导曲线，可以进一步了解它们的结构。

引用次数: 0

Non-adaptive prophet inequalities for minor-closed classes of matroids 拟阵小闭类的非自适应先知不等式

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-04-15 Epub Date: 2025-12-15 DOI: 10.1016/j.dam.2025.12.001

Kanstantsin Pashkovich, Alice Sayutina

We consider the matroid prophet inequality problem. This problem has been extensively studied in the case of adaptive mechanisms. In particular, there is a tight 2-competitive mechanism for all matroids (Kleinberg and Weinberg, 2012).

However, it is not known what classes of matroids admit non-adaptive mechanisms with constant guarantee. Recently, in Chawla et al. (2024) it was shown that there are constant-competitive non-adaptive mechanisms for graphic matroids. In this work, we show that various known classes of matroids admit constant-competitive non-adaptive mechanisms.

考虑一类拟阵的先知不等式问题。这个问题在自适应机制中得到了广泛的研究。特别是，所有拟阵都存在紧密的2-竞争机制（Kleinberg和Weinberg， 2012）。然而，尚不清楚哪类拟阵具有恒定保证的非自适应机制。最近，Chawla等人（2024）表明，图形拟阵存在持续竞争的非自适应机制。在这项工作中，我们证明了各种已知类拟阵承认不断竞争的非适应机制。

引用次数: 0

Even cycle decompositions of the line graphs of cubic graphs 三次图的线形图的连循环分解

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-04-15 Epub Date: 2026-01-09 DOI: 10.1016/j.dam.2025.12.052

Chuixiang Zhou, Yuxi Zou

An even cycle decomposition of a graph is defined as a partition of its edges into cycles of even length. Let

G

be a 2-connected cubic graph. Markström conjectured that the line graph

L (G)

G

admits an even cycle decomposition. Suppose that

F = C_{1} \cup C_{2} \cup \dots \cup C_{k}

is a 2-factor of

G

, where each

C_{i}

(

1 \leq i \leq k

) is a cycle. Let

G_{F}

be the graph obtained from

G

by contracting each

C_{i}

(

1 \leq i \leq k

) to a vertex and deleting the resulting loops and parallel edges. In this paper, we prove that Markström’s conjecture is true if

G_{F}

is isomorphic to a tree for some 2-factor

F

G

图的偶环分解被定义为将图的边划分为偶数长度的环。设G是一个二连通三次图。Markström推测G的线形图L(G)允许偶循环分解。设F=C1∪C2∪⋯∪Ck是G的一个2因子，其中每个Ci（1≤i≤k）是一个循环。设GF为由G将每个Ci（1≤i≤k）缩并到一个顶点，并删除由此产生的环路和平行边得到的图。本文证明了如果GF同构于某2因子F (G)的树，则Markström的猜想是成立的。

引用次数: 0

Non-inclusive g-extra diagnosability of interconnection networks under MM* model MM*模型下互联网络的非包容性g-extra可诊断性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-04-15 Epub Date: 2026-01-13 DOI: 10.1016/j.dam.2025.12.069

Weixing Zheng , Shuming Zhou , Eddie Cheng

With the popularization and deepening of applications on high-performance computing platforms, which are often built upon large-scale multiprocessor systems, system-level diagnosis has become essential to ensure system reliability and fault tolerability. Classical diagnosability and various conditional diagnosabilities are vital metrics to evaluate a system’s capability to accurately identify faulty processors. However, the upper bounds of these metrics commonly rely on the conventional assumptions of inclusiveness between faulty sets—one situation that rarely holds in practical scenarios. To overcome this deficiency, the non-inclusive diagnosability has been introduced and explored under various diagnostic models. Despite significant progress, the study of non-inclusive

g

-extra diagnosability under the MM* model remains up in the air. This work addresses this gap by determining the non-inclusive 2-extra diagnosability of general interconnection networks under the MM* model. As applications, we derive the non-inclusive 2-extra diagnosabilities of several well-known networks, including hypercube,

k

-ary

n

-cube, and bubble sort graph. In addition, we propose a novel diagnosis algorithm, NFDAM, for non-inclusive

g

-extra diagnosis, which runs in polynomial time with a complexity of

O (| V (G) | \cdot {(Δ (G))}^{3})

, where

Δ (G)

denotes the maximum degree of graph

G

. Simulation results demonstrate the effectiveness of the proposed algorithm in fault identification.

随着高性能计算平台应用的普及和深入，系统级诊断对于保证系统的可靠性和容错性至关重要。高性能计算平台通常建立在大型多处理器系统之上。经典可诊断性和各种条件可诊断性是评估系统准确识别故障处理器能力的重要指标。然而，这些指标的上限通常依赖于错误集之间的包容性的传统假设——这种情况在实际场景中很少成立。为了克服这一缺陷，在各种诊断模型下引入了非包容性可诊断性。尽管取得了重大进展，但在MM*模型下对非包容性g-extra可诊断性的研究仍然悬而未决。本工作通过确定MM*模型下一般互连网络的非包容性2-额外可诊断性来解决这一差距。作为应用，我们推导了几个著名网络的非包含2-额外可诊断性，包括超立方体、k-ary n-立方体和冒泡排序图。此外，我们提出了一种新的非包含G -extra诊断算法NFDAM，该算法运行时间为多项式，复杂度为O(|V(G)|⋅（Δ(G)）3)，其中Δ(G)表示图G的最大程度，仿真结果表明了该算法在故障识别中的有效性。

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引用次数: 0

On tree–wheel Ramsey numbers 在拉姆齐数字上

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-04-15 Epub Date: 2025-12-18 DOI: 10.1016/j.dam.2025.12.007

Yanbo Zhang , Yaojun Chen , Yunqing Zhang

Given two graphs

G

and

H

, the Ramsey number

R (G, H)

is the smallest positive integer

r

such that every graph on

r

vertices contains

G

as a subgraph or its complement contains

H

as a subgraph. Let

T_{n}

denote a tree on

n

vertices, and let

W_{m}

denote a wheel on

m + 1

vertices. Chen et al. (2004) asked for a characterization of those trees

T_{n}

satisfying

R (T_{n}, W_{2 m}) = 2 n - 1

for

m \geq 2

and

n \geq 2 m - 1

. Subsequently, Hafidh and Baskoro (2021) conjectured that if

T_{n}

is a tree with maximum degree at most

n - 2 m + 2

, then

R (T_{n}, W_{2 m}) = 2 n - 1

for

m \geq 2

and

n \geq 2 m + 1

. More recently, Britz et al. (2025) verified this conjecture partially for

m = 4

and large

n

In this note, we resolve the conjecture of Hafidh and Baskoro for all large

n

给定两个图G和H，拉姆齐数R（G，H）是最小的正整数R，使得在R个顶点上的每个图都包含G作为子图或其补包含H作为子图。Tn表示有n个顶点的树，Wm表示有m+1个顶点的轮。Chen等人（2004）要求对m≥2和n≥2m - 1时满足R(Tn,W2m)=2n - 1的树Tn进行表征。随后，Hafidh和Baskoro（2021）推测，如果Tn是最大度不超过n - 2m+2的树，则当m≥2和n≥2m+1时，R(Tn,W2m)=2n - 1。最近，Britz等人（2025）对m=4和大n部分验证了这一猜想。在本文中，我们解决了Hafidh和Baskoro对所有大n的猜想。

引用次数: 0

ILP and CP models for normalized integer weighted voting game design problem 正则化整数加权投票博弈设计问题的ILP和CP模型

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-03-31 Epub Date: 2025-12-04 DOI: 10.1016/j.dam.2025.11.051

Roger Kameugne , Pierre Wambo , Yves Pascal Ndjopnang Wantiep

Weighted voting games are a family of cooperative games, where each voter or player carries a weight. If the sum of the weights of the players in favor of a proposal is larger than or equal to a given quota, then this proposal is accepted. In the weighted game field, the problem of normalized integer weighted voting game representation (NIWVG), and the integer weighted voting game design problem (IWVGD) or inverse power index problem have been subjects of several studies. A solution to the last problem is an integer-weighted voting game with a specific power index, as close as possible (following a norm) to a given influence vector. Finding a normalized integer-weighted voting game solution for the inverse power index problem is a challenging task. The nature of the power index and the norm awarded in the problem can impact the approach used to solve this problem. In this paper, exact approaches based on integer linear programming (ILP) and constraint programming (CP) are proposed for this problem. The models proposed in this paper combine the Shapley–Shubik index or Banzhaf–Coleman index with Manhattan and Chebyshev norms respectively. Computational results show that ILP is an adequate approach when the power index and the norm can be linearized while CP appears to be suitable for the remaining cases of voting games with a small number of players.

加权投票游戏是一种合作游戏，其中每个投票人或玩家都有自己的权重。如果赞成某一提议的玩家的权重之和大于或等于给定的配额，则该提议被接受。在加权博弈领域，正则化整数加权投票博弈表示问题（NIWVG）和整数加权投票博弈设计问题（IWVGD）或幂指数逆问题已经成为许多研究的主题。最后一个问题的解决方案是使用具有特定权力指数的整数加权投票游戏，尽可能接近给定的影响向量（遵循规范）。幂指数反比问题的归一化整数加权投票博弈解是一项具有挑战性的任务。问题中功率指标的性质和授予的规范会影响解决该问题的方法。本文提出了基于整数线性规划（ILP）和约束规划（CP）的精确求解方法。本文提出的模型分别将Shapley-Shubik指数或Banzhaf-Coleman指数与Manhattan和Chebyshev范数相结合。计算结果表明，当幂指数和范数可以线性化时，ILP是一种合适的方法，而CP则适用于参与者较少的投票博弈的剩余情况。

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引用次数: 0

IP.LSH.DBSCAN: Integrated parallel density-based clustering by locality-sensitive hashing IP.LSH.DBSCAN：通过位置敏感散列集成并行基于密度的集群

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-03-31 Epub Date: 2025-12-01 DOI: 10.1016/j.dam.2025.11.047

Amir Keramatian, Vincenzo Gulisano, Marina Papatriantafilou, Philippas Tsigas

Locality-sensitive hashing (LSH) is an established method for fast data indexing and approximate similarity search, with useful parallelism properties. Although indexes and similarity measures are key for data clustering, little has been investigated on the multifaceted benefits of LSH in the problem. We show how approximate DBSCAN clustering can be fused into the process of creating an LSH index, and, through parallelization and fine-grained synchronization, also utilize efficiently available computing capacity. The resulting algorithm,

I P . L S H . D B S C A N

, described in this article, can support a wide range of applications with diverse distance functions, as well as data distributions and dimensionality. We analyse the algorithm’s asymptotic completion time and provide an open-source prototype implementation. We also conduct a detailed evaluation measuring latency and accuracy metrics of

I P . L S H . D B S C A N

, on a 36-core machine with 2-way hyper threading on massive data-sets with various numbers of dimensions. The analysis and the empirical study of

I P . L S H . D B S C A N

show how it complements the landscape of established state-of-the-art methods, by offering up to several orders of magnitude speed-up on higher dimensional datasets, with tunable high clustering accuracy.

位置敏感散列（LSH）是一种成熟的快速数据索引和近似相似性搜索方法，具有有用的并行性。虽然索引和相似性度量是数据聚类的关键，但很少有人研究LSH在问题中的多方面好处。我们将展示如何将近似DBSCAN聚类融合到创建LSH索引的过程中，并通过并行化和细粒度同步，有效地利用可用的计算能力。得到的算法IP.LSH。本文中描述的DBSCAN可以支持具有不同距离函数以及数据分布和维度的广泛应用程序。我们分析了算法的渐近完成时间，并提供了一个开源原型实现。我们还进行了详细的评估，测量IP.LSH的延迟和准确性指标。DBSCAN，在36核机器上使用双向超线程处理具有不同维数的大量数据集。IP.LSH.DBSCAN的分析和实证研究表明，通过在高维数据集上提供高达几个数量级的加速，并具有可调的高聚类精度，它如何补充了已建立的最先进的方法。

{"title":"IP.LSH.DBSCAN: Integrated parallel density-based clustering by locality-sensitive hashing","authors":"Amir Keramatian, Vincenzo Gulisano, Marina Papatriantafilou, Philippas Tsigas","doi":"10.1016/j.dam.2025.11.047","DOIUrl":"10.1016/j.dam.2025.11.047","url":null,"abstract":"<div><div>Locality-sensitive hashing (LSH) is an established method for fast data indexing and approximate similarity search, with useful parallelism properties. Although indexes and similarity measures are key for data clustering, little has been investigated on the multifaceted benefits of LSH in the problem. We show how approximate DBSCAN clustering can be fused into the process of creating an LSH index, and, through parallelization and fine-grained synchronization, also utilize efficiently available computing capacity. The resulting algorithm, <math><mstyle><mi>I</mi><mi>P</mi><mo>.</mo><mi>L</mi><mi>S</mi><mi>H</mi><mo>.</mo><mi>D</mi><mi>B</mi><mi>S</mi><mi>C</mi><mi>A</mi><mi>N</mi></mstyle></math>, described in this article, can support a wide range of applications with diverse distance functions, as well as data distributions and dimensionality. We analyse the algorithm’s asymptotic completion time and provide an open-source prototype implementation. We also conduct a detailed evaluation measuring latency and accuracy metrics of <math><mstyle><mi>I</mi><mi>P</mi><mo>.</mo><mi>L</mi><mi>S</mi><mi>H</mi><mo>.</mo><mi>D</mi><mi>B</mi><mi>S</mi><mi>C</mi><mi>A</mi><mi>N</mi></mstyle></math>, on a 36-core machine with 2-way hyper threading on massive data-sets with various numbers of dimensions. The analysis and the empirical study of <math><mstyle><mi>I</mi><mi>P</mi><mo>.</mo><mi>L</mi><mi>S</mi><mi>H</mi><mo>.</mo><mi>D</mi><mi>B</mi><mi>S</mi><mi>C</mi><mi>A</mi><mi>N</mi></mstyle></math> show how it complements the landscape of established state-of-the-art methods, by offering up to several orders of magnitude speed-up on higher dimensional datasets, with tunable high clustering accuracy.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 183-196"},"PeriodicalIF":1.0,"publicationDate":"2026-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Lines on digraphs of low diameter 低直径有向图上的线

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-03-31 Epub Date: 2025-12-13 DOI: 10.1016/j.dam.2025.12.015

Gabriela Araujo-Pardo , Martín Matamala , Juan Pablo Peña , José Zamora

A set of

n

non-collinear points in the Euclidean plane defines at least

n

different lines. Chen and Chvátal in 2008 conjectured that the same result is true in metric spaces for an adequate definition of line. More recently, it was conjectured in 2018 by Aboulker et al. that any large enough bridgeless graph on

n

vertices defines a metric space that has at least

n

lines.

We study the natural extension of Aboulker et al.’s conjecture into the context of quasi-metric spaces defined by digraphs of low diameter. We prove that it is valid for quasi-metric spaces defined by bipartite digraphs of diameter at most three, oriented graphs of diameter two and, digraphs of diameter three and directed girth four.

欧几里得平面上n个非共线点的集合定义了至少n条不同的直线。Chen和Chvátal在2008年推测，同样的结果在度量空间中也成立，因为有足够的线的定义。最近，Aboulker等人在2018年推测，任何足够大的n个顶点的无桥图都定义了一个至少有n条线的度量空间。我们研究了Aboulker等人的猜想在由低直径有向图定义的准度量空间中的自然推广。我们证明了它对于由直径最多为3的二部有向图、直径为2的有向图、直径为3的有向图和有向周长为4的有向图所定义的拟度量空间是有效的。

引用次数: 0

Note on two Kirchhoff index Turán problems 注意两个Kirchhoff指数Turán问题

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-03-31 Epub Date: 2025-11-21 DOI: 10.1016/j.dam.2025.11.007

Haiyan Chen , Wei Qiu

The Kirchhoff index of a connected graph is the sum of the resistance distance between all pairs of vertices. Bu et al. (Bounds on the resistance distance and Kirchhoff index of graphs, Discrete Appl. Math. 2025, 234-241) proved that (i) if

n

is divisible by

t

, then the Turán graph

T_{t} (n)

is the unique extremal graph with the minimum Kirchhoff index among all connected

K_{t + 1}

-free graphs with

n

vertices; (ii) if

n

is even, then the Turán graph

T_{2} (n)

is the unique extremal graph with the minimum Kirchhoff index among all connected

C_{2 t + 1}

-free graphs with

n

vertices

(n \geq 4 t - 2)

. They conjectured that the above two results hold for all

n

. In this note, we confirm their conjectures.

连通图的基尔霍夫指数是所有顶点对之间电阻距离的和。图的电阻距离和Kirchhoff指数的界限，离散应用。Math. 2025, 234-241)证明了(i)如果n能被t除，则Turán图Tt(n)是所有连通的Kt+1自由n顶点图中基尔霍夫指数最小的唯一极值图；（ii）如果n为偶数，则Turán图T2(n)是所有有n个顶点（n≥4t−2）的连通C2t+1自由图中基尔霍夫指数最小的唯一极值图。他们推测上述两个结果对所有n都成立。在本文中，我们证实了他们的推测。

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引用次数: 0

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