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Coulson-type integral formulas for the general (skew) Estrada index of a vertex 顶点一般（倾斜）埃斯特拉达指数的库尔森型积分公式

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-05 DOI: 10.1016/j.dam.2024.10.015

Lu Qiao , Shenggui Zhang , Jing Li , Nan Gao

<div><div>Let <span><math><mi>G</mi></math></span> be a simple graph with vertex set <span><math><mrow><mi>V</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> the eigenvalues of the adjacency matrix <span><math><mi>A</mi></math></span> of <span><math><mi>G</mi></math></span>. The Estrada index of <span><math><mi>G</mi></math></span> is defined as <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup></mrow></math></span>. The subgraph centrality of the vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> with respect to <span><math><mi>G</mi></math></span> is defined as the <span><math><mi>i</mi></math></span>th diagonal entry of the matrix <span><math><msup><mrow><mi>e</mi></mrow><mrow><mi>β</mi><mi>A</mi></mrow></msup></math></span>, where <span><math><mrow><mi>β</mi><mo>></mo><mn>0</mn></mrow></math></span>. Let <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup></math></span> be the oriented graph of <span><math><mi>G</mi></math></span> with an orientation <span><math><mi>σ</mi></math></span> and <span><math><mrow><msub><mrow><mi>ζ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> the eigenvalues of the skew-adjacency matrix of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup></math></span>. The skew Estrada index of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup></math></span> is defined as <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>ζ</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup></mrow></math></span>. Gao et al. obtained some Coulson-type integral formulas for the Estrada index of <span><math><mi>G</mi></math></span> and for the skew Estrada index of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>σ</mi></mrow></msup></math></span>. In this paper, we will introduce the concept of the general Estrada index of <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> with respect to <span><math><mi>G</mi></math></span> as a

设 G 为简单图，顶点集 V={v1,v2,...,vn}，λ1,λ2,...,λn 为 G 的邻接矩阵 A 的特征值。顶点 vi 相对于 G 的子图中心度定义为矩阵 eβA 的第 β 个对角项，其中 β>0。假设 Gσ 是方向为 σ 的 G 的有向图，ζ1,ζ2,...,ζn 是 Gσ 的倾斜-相接矩阵的特征值。Gσ 的偏斜埃斯特拉达指数定义为 ∑k=1neiζk。Gao 等人得到了一些关于 G 的埃斯特拉达指数和 Gσ 的偏斜埃斯特拉达指数的库尔森型积分公式。本文将介绍作为子图中心性广义化的vi相对于G的广义Estrada指数的概念和vi相对于Gσ的广义偏斜Estrada指数的概念，并给出G的广义顶点Estrada指数和Gσ的广义顶点偏斜Estrada指数的一些库伦式积分公式。

引用次数: 0

Efficient enumeration of transversal edge-partitions 高效枚举横向边缘分区

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-04 DOI: 10.1016/j.dam.2024.10.016

Koki Shinraku, Katsuhisa Yamanaka, Takashi Hirayama

An irreducible triangulation is a plane graph such that its outer face is a quadrangle, every inner face is a triangle, and it has no separating triangle. Let

T

be an irreducible triangulation with

n

vertices. A rectangular dual

R

T

is a dissection of a rectangle into (small) rectangles such that (1) each rectangle of

R

corresponds to a vertex of

T

, and (2) two rectangles of

R

are adjacent if the two corresponding vertices of

T

are adjacent. Finding a rectangular dual of a given graph has an application on cartograms and VLSI floor-planning. In this paper, we consider the problem of enumerating all the rectangular duals of a given irreducible triangulation. It is known that the set of rectangular duals of an irreducible triangulation

T

one-to-one corresponds to the set of transversal edge-partitions of

T

. Hence, in this paper, we design an enumeration algorithm of all the transversal edge-partitions of an irreducible triangulation with

n

vertices. The proposed algorithm enumerates them in

O (n)

-delay and

O (n^{2})

-space after

O (n log n)

-time preprocessing.

不可还原三角形是这样一个平面图形：它的外侧面是一个四边形，每个内侧面都是一个三角形，并且没有分离三角形。设 T 是一个有 n 个顶点的不可还原三角形。T 的矩形对偶 R 是将矩形分割成（小）矩形，这样 (1) R 的每个矩形都对应 T 的一个顶点；(2) 如果 T 的两个对应顶点相邻，则 R 的两个矩形相邻。寻找给定图形的矩形对偶可应用于制图和超大规模集成电路平面规划。在本文中，我们考虑的问题是枚举给定不可还原三角形的所有矩形对偶。众所周知，不可还原三角形 T 的矩形对偶集一一对应于 T 的横向边分区集。因此，在本文中，我们设计了一种枚举具有 n 个顶点的不可还原三角形的所有横向边分区的算法。经过 O(nlogn)-time 的预处理后，所提出的算法能在 O(n)-delay 和 O(n2)-space 内枚举出它们。

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引用次数: 0

Efficient constant-factor approximate enumeration of minimal subsets for monotone properties with weight constraints 带权重约束的单调属性最小子集的高效恒因子近似枚举

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.dam.2024.10.014

Yasuaki Kobayashi , Kazuhiro Kurita , Kunihiro Wasa

A property

Π

on a finite set

U

is monotone if for every

X \subseteq U

satisfying

Π

, every superset

Y \subseteq U

X

also satisfies

Π

. Many combinatorial properties can be seen as monotone properties. The problem of finding a subset of

U

satisfying

Π

with the minimum weight is a central problem in combinatorial optimization. Although many approximate/exact algorithms have been developed to solve this kind of problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building accurate mathematical models on real-world problems. A promising approach to overcome this difficulty is to enumerate multiple small solutions rather than to find a single small solution. To this end, given a weight function

w : U \to Q_{> 0}

and

k \in Q_{> 0}

, we devise algorithms that approximately enumerate all minimal subsets of

U

with weight at most

k

satisfying

Π

for various monotone properties

Π

, where “approximate enumeration” means that algorithms output all minimal subsets satisfying

Π

whose weight is at most

k

and may output some minimal subsets satisfying

Π

whose weight exceeds

k

but is at most

c k

for some constant

c \geq 1

. These algorithms allow us to efficiently enumerate minimal vertex covers, minimal dominating sets in bounded degree graphs, minimal feedback vertex sets, minimal hitting sets in bounded rank hypergraphs, etc., of weight at most

k

with constant approximation factors.

如果对于满足 Π 的每个 X⊆U，X 的每个超集 Y⊆U 也满足 Π，那么有限集 U 上的属性 Π 就是单调的。许多组合性质都可以看作是单调性质。以最小权重找到满足 Π 的 U 子集是组合优化的核心问题。虽然已经开发出了许多近似/精确算法来解决这类问题，但由于很难针对实际问题建立精确的数学模型，这些算法得到的解决方案往往不适合实际应用。克服这一困难的一种可行方法是枚举多个小解决方案，而不是寻找单一的小解决方案。为此，给定一个权重函数 w:U→Q>0 和 k∈Q>0，我们设计了一些算法，可以近似枚举权重至多为 k、满足各种单调属性 Π 的 U 的所有最小子集，这里的 "近似枚举 "是指算法输出所有满足 Π 的最小子集，其权重至多为 k，并且可能输出一些满足 Π 的最小子集，其权重超过 k，但在某个常数 c≥1 时至多为 ck。通过这些算法，我们可以高效地枚举最小顶点覆盖、有界度图中的最小支配集、最小反馈顶点集、有界秩超图中的最小命中集等、权重最多为 k，且近似系数恒定。

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引用次数: 0

Quasi-kernels in split graphs 分裂图中的准核

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.dam.2024.10.009

Hélène Langlois , Frédéric Meunier , Romeo Rizzi , Stéphane Vialette , Yacong Zhou

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that the shortest path from every vertex to this subset is of length at most two. The “small quasi-kernel conjecture”, proposed by Erdős and Székely in 1976, postulates that every sink-free digraph has a quasi-kernel whose size is within a fraction of the total number of vertices. The conjecture is even more precise with a

1 / 2

ratio, but even with larger ratio, this property is known to hold only for few classes of graphs.

The focus here is on small quasi-kernels in split graphs. This family of graphs has played a special role in the study of the conjecture since it was used to disprove a strengthening that postulated the existence of two disjoint quasi-kernels. The paper proves that every sink-free split digraph

D

has a quasi-kernel of size at most

\frac{2}{3} | V (D) |

, and even of size at most two when the graph is an orientation of a complete split graph. It is also shown that computing a quasi-kernel of minimal size in a split digraph is W[2]-hard.

在一个数图中，准核是一个独立的顶点子集，使得从每个顶点到这个子集的最短路径的长度最多为 2。Erdős 和 Székely 于 1976 年提出了 "小准核猜想"，假设每个无汇数图都有一个准核，其大小在顶点总数的几分之一以内。这一猜想在比率为 1/2 时更为精确，但即使比率更大，这一性质也只在少数几类图中成立。这个图族在该猜想的研究中发挥了特殊作用，因为它被用来推翻假设存在两个不相交准核的强化。本文证明了每个无汇分裂数图 D 最多有一个大小为 23|V(D)| 的准核，当该图是一个完整分裂图的定向时，甚至最多有一个大小为两个的准核。同时还证明，计算一个分裂数图中最小尺寸的准核是 W[2]-hard 的。

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引用次数: 0

Intersection of chordal graphs and some related partition problems 弦图的相交和一些相关的分割问题

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.dam.2024.10.010

Atif Abueida , Arthur Busch , R. Sritharan

The chordality of a graph is the minimum number of chordal graphs whose intersection is the graph. A result of Yannakakis’ from 1982 can be used to infer that for every fixed

k \geq 3

, deciding whether the chordality of a graph is at most

k

is NP-complete. We consider the problem of deciding whether the chordality of a graph is 2, or equivalently, deciding whether a given graph is the intersection of two chordal graphs. We prove that the problem is equivalent to a partition problem when one of the chordal graphs is a split graph and the other meets certain conditions. Using this we derive complexity results for a variety of problems, including deciding if a graph is the intersection of

k

split graphs, which is in P for

k = 2

and NP-complete for

k \geq 3

一个图的和弦度是其交集为该图的和弦图的最小数目。可以利用扬纳卡基斯（Yannakakis）1982 年的一个结果来推断，对于每个固定的 k≥3，判断一个图的和弦度是否最多为 k 是 NP-complete。我们考虑的问题是判定一个图的和弦度是否为 2，或者等价于判定一个给定的图是否是两个和弦图的交集。我们证明，当其中一个弦图是分裂图，而另一个满足特定条件时，该问题等同于分割问题。利用这一点，我们推导出了各种问题的复杂性结果，包括判定一个图是否是 k 个分裂图的交集，对于 k=2 的问题，该问题在 P 级，而对于 k≥3 的问题，该问题在 NP 级。

引用次数: 0

Pure Nash equilibria in weighted matroid congestion games with non-additive aggregation and beyond 非加权聚合加权矩阵拥塞博弈中的纯纳什均衡及其他

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.dam.2024.10.017

Kenjiro Takazawa

Congestion games offer a primary model in the study of pure Nash equilibria in non-cooperative games, and a number of generalized models have been proposed in the literature. One line of generalization includes weighted congestion games, in which the cost of a resource is a function of the total weight of the players choosing that resource. Another line includes congestion games with mixed costs, in which the cost imposed on a player is a convex combination of the total cost and the maximum cost of the resources in her strategy. This model is further generalized to that of congestion games with non-additive aggregation. For the above models, the existence of a pure Nash equilibrium is proved under some assumptions, including the case in which the strategy space of each player is the base family of a matroid and the case in which the cost functions have a certain kind of monotonicity. In this paper, we deal with common generalizations of these two lines, namely weighted matroid congestion games with non-additive aggregation, and its further generalization. Our main technical contribution is a proof of the existence of pure Nash equilibria in these generalized models under a simplified assumption on the monotonicity, which provides a common extension of the previous results. We also present an extension on the existence of pure Nash equilibria in weighted matroid congestion games with mixed costs.

拥挤博弈是研究非合作博弈中纯纳什均衡的一个主要模型，文献中也提出了许多广义模型。其中一种广义模型包括加权拥挤博弈，在这种博弈中，一种资源的成本是选择该资源的博弈者总权重的函数。另一个模型包括混合成本的拥堵博弈，在这种博弈中，棋手的成本是其策略中资源的总成本和最大成本的凸组合。这一模型被进一步归纳为非加总的拥挤博弈模型。对于上述模型，我们在一些假设条件下证明了纯纳什均衡的存在，包括每个博弈方的策略空间是一个矩阵的基族，以及成本函数具有某种单调性。在本文中，我们将讨论这两种情况的共同概括，即具有非加性聚合的加权矩阵拥塞博弈及其进一步概括。我们的主要技术贡献是在简化的单调性假设下证明了这些广义模型中纯纳什均衡的存在性，这是对之前结果的共同扩展。我们还对具有混合成本的加权矩阵拥塞博弈中纯纳什均衡的存在性进行了扩展。

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引用次数: 0

On the eccentric connectivity index of trees with given domination number 关于具有给定支配数的树木的偏心连通指数

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.dam.2024.10.013

Ting Zhou , Lianying Miao , Zhen Lin , Wenyao Song

Let

G

be a simple connected finite graph. The eccentric connectivity index (ECI) of

G

is defined as

ξ^{c} (G) = \sum_{v \in V (G)} ɛ_{G} (v) d_{G} (v)

, where

ɛ_{G} (v)

is the eccentricity of

v

d_{G} (v)

is the degree of

v

. We denote the set of trees with order

n

and domination number

γ

T_{n, γ}

. In this paper, the extremal trees having the minimal ECI among

T_{n, γ}

are determined. The tree among

T_{n, γ}

satisfying

2 \leq γ \leq ⌈ \frac{n}{3} ⌉

having the maximal ECI is also characterized. For

⌈ \frac{n}{3} ⌉ \leq γ \leq \frac{n}{2}

, the tree among all caterpillars with domination number

γ

having the maximal ECI is determined.

设 G 是一个简单连通的有限图。G 的偏心连通指数（ECI）定义为ξc(G)=∑v∈V(G)ɛG(v)dG(v)，其中ɛG(v)是 v 的偏心率，dG(v) 是 v 的度数。我们用 Tn,γ 表示阶数为 n、主宰数为 γ 的树集合。本文将确定 Tn,γ 中 ECI 最小的极值树。此外，本文还描述了 Tn,γ 中满足 2≤γ≤⌈n3⌉ 且具有最大 ECI 的树。对于⌈n3⌉≤γ≤n2，确定了具有最大 ECI 的支配数 γ 的所有毛虫中的树。

引用次数: 0

Re-1-embeddings of optimal 1-embedded graphs on the projective plane 投影面上最优 1 嵌入图的 Re-1 嵌入

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.dam.2024.10.005

Yusuke Suzuki

It was shown in Schumacher (1986), Suzuki (2010) that every optimal 1-embedded graph on the sphere has at most 8 inequivalent 1-embeddings. In this paper, we prove that the number of inequivalent 1-embeddings of an optimal 1-embedded graph on the projective plane whose quadrangular subgraph is bipartite is at most 24. In the case where such quadrangular subgraphs are nonbipartite, we show an optimal 1-embedded graph having at least

p

inequivalent 1-embeddings for any large integer

p

Schumacher (1986) 和 Suzuki (2010) 的研究表明，球面上的每个最优 1 嵌入图最多有 8 个不等价 1 嵌入。在本文中，我们证明了投影平面上的最优 1- 嵌入图的不等价 1- 嵌入的数量最多为 24，而该图的四角子图是双方形的。在这种四角形子图是非双方形的情况下，我们证明了一个最优 1 嵌入图在任意大整数 p 下至少有 p 个不等价 1 嵌入。

引用次数: 0

Self-adhesivity in lattices of abstract conditional independence models 抽象条件独立模型网格中的自粘性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.dam.2024.10.006

Tobias Boege , Janneke H. Bolt , Milan Studený

We introduce an algebraic concept of the frame for abstract conditional independence (CI) models, together with basic operations with respect to which such a frame should be closed: copying and marginalization. Three standard examples of such frames are (discrete) probabilistic CI structures, semi-graphoids and structural semi-graphoids. We concentrate on those frames which are closed under the operation of set-theoretical intersection because, for these, the respective families of CI models are lattices. This allows one to apply the results from lattice theory and formal concept analysis to describe such families in terms of implications among CI statements.

The central concept of this paper is that of self-adhesivity defined in algebraic terms, which is a combinatorial reflection of the self-adhesivity concept studied earlier in context of polymatroids and information theory. The generalization also leads to a self-adhesivity operator defined on the meta-level of CI frames. We answer some of the questions related to this approach and raise other open questions.

The core of the paper is in computations. The combinatorial approach to computation might overcome some memory and space limitation of software packages based on polyhedral geometry, in particular, if SAT solvers are utilized. We characterize some basic CI families over 4 variables in terms of canonical implications among CI statements. We apply our method in information-theoretical context to the task of entropic region demarcation over 5 variables.

我们为抽象条件独立性（CI）模型引入了框架的代数概念，并介绍了这种框架应该封闭的基本操作：复制和边际化。这种框架的三个标准例子是（离散）概率 CI 结构、半图形和结构半图形。我们将重点放在那些在集合论交集操作下封闭的框架上，因为对于这些框架来说，各自的 CI 模型族都是网格。本文的核心概念是用代数术语定义的自粘性，它是早先在多面体和信息论背景下研究的自粘性概念的组合反映。这一概括也导致了在 CI 框架元层面上定义的自粘性算子。我们回答了与这种方法相关的一些问题，并提出了其他开放性问题。计算的组合方法可以克服基于多面体几何的软件包在内存和空间方面的限制，特别是在使用 SAT 求解器的情况下。我们用 CI 语句之间的典型含义来描述一些基本的 4 变量 CI 族。我们在信息论背景下将我们的方法应用于 5 个变量的熵区域划分任务。

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引用次数: 0

On the chromatic number of powers of subdivisions of graphs 论图形细分幂的色度数

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-10-23 DOI: 10.1016/j.dam.2024.10.002

Michael Anastos , Simona Boyadzhiyska , Silas Rathke , Juanjo Rué

For a given graph

G = (V, E)

, we define its

n

th subdivision as the graph obtained from

G

by replacing every edge by a path of length

n

. We also define the

m

th power of

G

as the graph on vertex set

V

where we connect every pair of vertices at distance at most

m

G

. In this paper, we study the chromatic number of powers of subdivisions of graphs and resolve the case

m = n

asymptotically. In particular, our result confirms a conjecture of Mozafari-Nia and Iradmusa in the case

m = n = 3

in a strong sense.

对于给定的图 G=(V,E)，我们将其第 n 次细分定义为将每条边替换为长度为 n 的路径后从 G 中得到的图。我们还将 G 的第 m 次幂定义为顶点集 V 上的图，在该图中，我们以最多 m 的距离连接 G 中的每对顶点。特别是，我们的结果在强意义上证实了 Mozafari-Nia 和 Iradmusa 在 m=n=3 情况下的猜想。

引用次数: 0

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