Discrete Mathematics最新文献_第8页

Synchronicity of descent and excedance enumerators in the alternating subgroup 交替子群中下降和超越枚举数的同步性

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-07 DOI: 10.1016/j.disc.2025.114521

Umesh Shankar

引用次数: 0

The Li-Feng transformation of weighted adjacency matrices for graphs with degree-based edge-weights 基于边权重度的图的加权邻接矩阵的李丰变换

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-07 DOI: 10.1016/j.disc.2025.114520

Jing Gao, Xueliang Li, Ning Yang, Ruiling Zheng

{"title":"The Li-Feng transformation of weighted adjacency matrices for graphs with degree-based edge-weights","authors":"Jing Gao, Xueliang Li, Ning Yang, Ruiling Zheng","doi":"10.1016/j.disc.2025.114520","DOIUrl":"10.1016/j.disc.2025.114520","url":null,"abstract":"<div><div>For a graph G, let <math><msub><mrow><mi>d</mi></mrow><mrow><mi>v</mi></mrow></msub></math> be the degree of a vertex v. Given a symmetric real function <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math>, the weight of edge uv in graph G is equal to the value <math><mi>f</mi><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>)</mo></math>. The degree-based weighted adjacency matrix is defined as <math><msub><mrow><mi>A</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math>, in which the <math><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></math>-entry is equal to <math><mi>f</mi><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>)</mo></math> if uv is an edge of G and 0 otherwise. In this paper, we consider the Li-Feng transformation and show that if a graph G contains two pendant paths on a common vertex, the uniform distribution of pendant paths increases the largest eigenvalue of <math><msub><mrow><mi>A</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math>, when <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math> is increasing in x and the length of two pendant paths should be at least 2. We also consider the cycle version of Li-Feng transformation and show that if a graph G contains two pendant cycles on a common vertex, the uniform distribution of pendant cycles decreases the largest eigenvalue of <math><msub><mrow><mi>A</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math>, when <math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>></mo><mn>2</mn><mi>f</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>. The purpose of this paper is to unify the study of the graph operation on the largest eigenvalue for the degree-based weighted adjacency matrix.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114520"},"PeriodicalIF":0.7,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143786302","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

Interlacing property of polynomial sequences related to multinomial coefficients 与多项系数相关的多项式序列的隔行性

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-05 DOI: 10.1016/j.disc.2025.114522

Ming-Jian Ding

引用次数: 0

Bipartite binding number, k-factor and spectral radius of bipartite graphs 二方图的二方结合数、k 因子和谱半径

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-05 DOI: 10.1016/j.disc.2025.114511

Yifang Hao , Shuchao Li , Yuantian Yu

引用次数: 0

Exploring the influence of graph operations on zero forcing sets 探讨图运算对零强迫集的影响

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-05 DOI: 10.1016/j.disc.2025.114516

Krishna Menon , Anurag Singh

{"title":"Exploring the influence of graph operations on zero forcing sets","authors":"Krishna Menon , Anurag Singh","doi":"10.1016/j.disc.2025.114516","DOIUrl":"10.1016/j.disc.2025.114516","url":null,"abstract":"<div><div>Zero forcing in graphs is a coloring process where a vertex colored blue can force its unique uncolored neighbor to be colored blue. A zero forcing set is a set of initially blue vertices capable of eventually coloring all vertices of the graph. In this paper, we focus on the numbers <math><mi>z</mi><mo>(</mo><mi>G</mi><mo>;</mo><mi>i</mi><mo>)</mo></math>, which is the number of zero forcing sets of size i of the graph G. These numbers were initially studied by Boyer et al. [5] where they conjectured that for any graph G on n vertices, <math><mi>z</mi><mo>(</mo><mi>G</mi><mo>;</mo><mi>i</mi><mo>)</mo><mo>≤</mo><mi>z</mi><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><mi>i</mi><mo>)</mo></math> for all <math><mi>i</mi><mo>≥</mo><mn>1</mn></math> where <math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is the path graph on n vertices. The main aim of this paper is to show that several classes of graphs, including outerplanar graphs and threshold graphs, satisfy this conjecture. We do this by studying various graph operations and examining how they affect the number of zero forcing sets.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114516"},"PeriodicalIF":0.7,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On bipartite graphs with the minimum number of spanning trees 关于具有最少生成树的二叉图

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-03 DOI: 10.1016/j.disc.2025.114514

Shicai Gong, Yue Xu, Peng Zou, Jiaxin Wang

{"title":"On bipartite graphs with the minimum number of spanning trees","authors":"Shicai Gong, Yue Xu, Peng Zou, Jiaxin Wang","doi":"10.1016/j.disc.2025.114514","DOIUrl":"10.1016/j.disc.2025.114514","url":null,"abstract":"<div><div>The collection of all (simple and connected) bipartite graphs with cyclomatic number ω is denoted by <math><msub><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msub></math>. We use <math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>;</mo><mi>b</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math> to denote the graph obtained from the complete bipartite graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>+</mo><mn>1</mn></mrow></msub></math> by removing <math><mi>a</mi><mo>−</mo><mi>c</mi></math> edges that are all connected to the same vertex of degree a, here <math><mi>a</mi><mo>,</mo><mi>b</mi></math> and c are integers with <math><mn>2</mn><mo>≤</mo><mi>c</mi><mo><</mo><mi>a</mi><mo>≤</mo><mi>b</mi></math>. The term <math><mi>S</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> denotes the skeleton of the graph G, which is defined as the largest induced subgraph of G that contains no pendant vertices.</div><div>In this paper, we investigate the problem of characterizing the graphs within <math><msub><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msub></math> that possess the minimum number of spanning trees. We show that the skeleton of each graph with the minimum number of spanning trees in <math><msub><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msub></math> is either <math><msub><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub></math>, where a and b are positive integers with <math><mn>2</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>b</mi></math> and <math><mo>(</mo><mi>a</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>ω</mi></math>, or <math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>;</mo><mi>b</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math>, where <math><mi>a</mi><mo>,</mo><mi>b</mi></math> and c are positive integers satisfying <math><mn>2</mn><mo>≤</mo><mi>c</mi><mo><</mo><mi>a</mi><mo>≤</mo><mi>b</mi></math> and <math><mi>c</mi><mo>−</mo><mn>1</mn><mo>+</mo><mo>(</mo><mi>a</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>ω</mi></math>. In addition, we establish some structural properties by the method of analysis to further reduce those candidate graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114514"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759069","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

On the maximum number of r-cliques in graphs free of complete r-partite subgraphs 无完全r部子图的图中r-团的最大数目

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-03 DOI: 10.1016/j.disc.2025.114508

József Balogh , Suyun Jiang , Haoran Luo

引用次数: 0

The saturation number of C6 C6的饱和数

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-03 DOI: 10.1016/j.disc.2025.114504

Yongxin Lan , Yongtang Shi , Yiqiao Wang , Junxue Zhang

引用次数: 0

Every even cycle of order at least 8 has a mirror labeling 每一个至少为8阶的偶数循环都有一个镜像标记

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-03 DOI: 10.1016/j.disc.2025.114503

Jonathan Calzadillas , Dan McQuillan , James M. McQuillan

{"title":"Every even cycle of order at least 8 has a mirror labeling","authors":"Jonathan Calzadillas , Dan McQuillan , James M. McQuillan","doi":"10.1016/j.disc.2025.114503","DOIUrl":"10.1016/j.disc.2025.114503","url":null,"abstract":"<div><div>A mirror labeling of the cycle <math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is a vertex-magic total labeling (VMTL) for the cycle <math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math> with the property that if x is a vertex label, then <math><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>x</mi></math> is an edge label, for each <math><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn><mi>n</mi></math>. (Note that any mirror labeling for <math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math> can be easily converted into an edge-magic total labeling for <math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math> with the same property, and vice versa.) It has been known for decades that every odd cycle has a mirror labeling. Mirror labelings for <math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math> with n even are considerably more difficult to construct generally, with only the case <math><mi>n</mi><mo>≡</mo><mn>2</mn></math> mod 8 having been provided. In this paper, we obtain mirror labelings for all remaining cases, namely <math><mi>n</mi><mo>≡</mo><mn>6</mn></math> mod 8, <math><mi>n</mi><mo>≥</mo><mn>14</mn></math> and <math><mi>n</mi><mo>≡</mo><mn>0</mn></math> mod 4, <math><mi>n</mi><mo>≥</mo><mn>8</mn></math>.</div><div>This result has significant ramifications for the study of vertex-magic total labelings of graphs generally. A quarter century ago, James MacDougall provided his guiding conjecture positing that every regular graph of degree at least 2 has a VMTL, except for the disjoint union <math><mn>2</mn><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math>. Ian Gray showed that every Hamiltonian regular graph of odd order possesses a VMTL, and introduced mirror vertex-magic total labelings as a tool to obtain a similar, general result for even order regular graphs. However, a key technical part of his program was missing, namely, the existence of mirror VMTLs for even order cycles. A mirror labeling is a particular kind of mirror VMTL. Thus, the results of this work provide the missing piece required for Gray's program. It now follows, that any Hamiltonian <math><mo>(</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>)</mo></math>-regular graph of any even order (<math><mi>n</mi><mo>≥</mo><mn>8</mn></math>, <math><mi>t</mi><mo>≥</mo><mn>0</mn></math>) must have a VMTL. This provides substantial new progress towards resolving MacDougall's Conjecture.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114503"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761184","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

Rainbow directed version of Dirac's theorem 狄拉克定理的彩虹定向版本

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-04-03 DOI: 10.1016/j.disc.2025.114506

Hao Li , Luyi Li , Ping Li , Xueliang Li

{"title":"Rainbow directed version of Dirac's theorem","authors":"Hao Li , Luyi Li , Ping Li , Xueliang Li","doi":"10.1016/j.disc.2025.114506","DOIUrl":"10.1016/j.disc.2025.114506","url":null,"abstract":"<div><div>Let <math><mi>G</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mo>[</mo><mi>s</mi><mo>]</mo><mo>}</mo></math> be a collection of not necessarily distinct graphs on the same vertex set V. A graph H is called rainbow in <math><mi>G</mi></math> if any two edges of H belong to different graphs of <math><mi>G</mi></math>. In 2020, Joos and Kim proved a rainbow version of Dirac's theorem. In this paper, we prove a rainbow directed version of Dirac's theorem asymptotically: For each <math><mn>0</mn><mo><</mo><mi>ε</mi><mo><</mo><mn>1</mn></math>, there exists an integer N such that when <math><mi>n</mi><mo>≥</mo><mi>N</mi></math> the following holds. Let <math><mi>D</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>]</mo><mo>}</mo></math> be a collection of n-vertex digraphs on the same vertex set V. If both the out-degree and the in-degree of v are at least <math><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ε</mi><mo>)</mo></mrow><mi>n</mi></math> for each vertex <math><mi>v</mi><mo>∈</mo><mi>V</mi></math> and each integer <math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>]</mo></math>, then <math><mi>D</mi></math> contains a rainbow Hamiltonian cycle. Furthermore, we provide a sufficient condition for the existence of arbitrary rainbow tournaments in a collection of n-vertex digraphs, where a tournament is an oriented graph of a complete graph.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114506"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761335","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