Discrete Mathematics最新文献

{"title":"Distance-regular graphs with classical parameters that support a uniform structure: Case q ≥ 2","authors":"","doi":"10.1016/j.disc.2024.114263","DOIUrl":"10.1016/j.disc.2024.114263","url":null,"abstract":"<div><p>Let <span><math><mi>Γ</mi><mo>=</mo><mo>(</mo><mi>X</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> denote a finite, simple, connected, and undirected non-bipartite graph with vertex set <em>X</em> and edge set <span><math><mi>R</mi></math></span>. Fix a vertex <span><math><mi>x</mi><mo>∈</mo><mi>X</mi></math></span>, and define <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mi>R</mi><mo>∖</mo><mo>{</mo><mi>y</mi><mi>z</mi><mo>|</mo><mo>∂</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mo>∂</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>)</mo><mo>}</mo></math></span>, where ∂ denotes the path-length distance in Γ. Observe that the graph <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>)</mo></math></span> is bipartite. We say that Γ supports a uniform structure with respect to <em>x</em> whenever <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>f</mi></mrow></msub></math></span> has a uniform structure with respect to <em>x</em> in the sense of Miklavič and Terwilliger <span><span>[7]</span></span>.</p><p>Assume that Γ is a distance-regular graph with classical parameters <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span> and diameter <span><math><mi>D</mi><mo>≥</mo><mn>4</mn></math></span>. Recall that <em>q</em> is an integer such that <span><math><mi>q</mi><mo>∉</mo><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>}</mo></math></span>. The purpose of this paper is to study when Γ supports a uniform structure with respect to <em>x</em>. We studied the case <span><math><mi>q</mi><mo>≤</mo><mn>1</mn></math></span> in <span><span>[3]</span></span>, and so in this paper we assume <span><math><mi>q</mi><mo>≥</mo><mn>2</mn></math></span>. Let <span><math><mi>T</mi><mo>=</mo><mi>T</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> denote the Terwilliger algebra of Γ with respect to <em>x</em>. Under an additional assumption that every irreducible <em>T</em>-module with endpoint 1 is thin, we show that if Γ supports a uniform structure with respect to <em>x</em>, then either <span><math><mi>α</mi><mo>=</mo><mn>0</mn></math></span> or <span><math><mi>α</mi><mo>=</mo><mi>q</mi></math></span>, <span><math><mi>β</mi><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>D</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>, and <span><math><mi>D</mi><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>6</mn><mo>)</mo></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003947/pdfft?md5=1365ed5c25a5773efbf51cb8def0b01e&pid=1-s2.0-S0012365X24003947-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142238249","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}

{"title":"New results on asymmetric orthogonal arrays with strength t ≥ 3","authors":"","doi":"10.1016/j.disc.2024.114264","DOIUrl":"10.1016/j.disc.2024.114264","url":null,"abstract":"<div><p>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 <span><math><mi>t</mi><mo>≥</mo><mn>3</mn></math></span> are obtained. Furthermore, some new infinite families of large sets of orthogonal arrays with mixed levels are also obtained.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142242252","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}

{"title":"The weight hierarchies of linear codes from simplicial complexes","authors":"","doi":"10.1016/j.disc.2024.114240","DOIUrl":"10.1016/j.disc.2024.114240","url":null,"abstract":"<div><p>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 <span><math><mi>C</mi></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, which is constructed from defining sets. These defining sets are either special simplicial complexes or their complements in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span>. We determine the complete weight hierarchies of these codes by analyzing the maximum or minimum intersection of certain simplicial complexes and all <em>r</em>-dimensional subspaces of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span>, where <span><math><mn>1</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><msub><mrow><mi>dim</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003716/pdfft?md5=b31c93fc7520c0f919446480a13b7f62&pid=1-s2.0-S0012365X24003716-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173918","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}

{"title":"On the restricted order of asymptotic bases","authors":"","doi":"10.1016/j.disc.2024.114260","DOIUrl":"10.1016/j.disc.2024.114260","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be the set of all positive integers. For a set &lt;em&gt;A&lt;/em&gt; of positive integers, let &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; denote that &lt;em&gt;A&lt;/em&gt; contains all but finitely many positive integers. For an integer &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;⩾&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, define &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⋯&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. In 2023, Chen and Yu [Discrete Math. 346 (2023), Paper No. 113388.] proved that, there exists a set &lt;em&gt;B&lt;/em&gt; of positive integers such that: &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;⋃&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;⋃&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≁&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;⋃&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⋃&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we construct a &lt;em&gt;somewhat dense&lt;/em&gt; set &lt;em&gt;B&lt;/em&gt; satisfying the above properties. That is, there exists a set &lt;em&gt;B&lt;/em&gt; of positive integers such that: &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;inf&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;sup&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;⋃&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;⋃&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;m","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003911/pdfft?md5=aacfc54f27829de05568c6d3ed5aa0a2&pid=1-s2.0-S0012365X24003911-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173812","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}

{"title":"Oriented posets, rank matrices and q-deformed Markov numbers","authors":"","doi":"10.1016/j.disc.2024.114256","DOIUrl":"10.1016/j.disc.2024.114256","url":null,"abstract":"<div><p>We define <em>oriented posets</em> with corresponding <em>rank matrices</em>, 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 <em>q</em>-deformed Markov numbers. We also resolve a conjecture of Leclere and Morier-Genoud and give several identities between circular rank polynomials.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142172371","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}

{"title":"Diversity and intersecting theorems for weak compositions","authors":"","doi":"10.1016/j.disc.2024.114250","DOIUrl":"10.1016/j.disc.2024.114250","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> be the set of non-negative integers, and let <span><math><mi>P</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> denote the set of all weak compositions of <em>n</em> with <em>k</em> parts, i.e., <span><math><mi>P</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo><mo>=</mo><mo>{</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>∈</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mspace></mspace><mo>:</mo><mspace></mspace><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>}</mo></math></span>. For any element <span><math><mi>u</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>P</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span>, denote its <em>i</em>th-coordinate by <span><math><mi>u</mi><mo>(</mo><mi>i</mi><mo>)</mo></math></span>, i.e., <span><math><mi>u</mi><mo>(</mo><mi>i</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. A family <span><math><mi>A</mi><mo>⊆</mo><mi>P</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> is said to be <em>t</em>-intersecting if <span><math><mo>|</mo><mo>{</mo><mi>i</mi><mspace></mspace><mo>:</mo><mspace></mspace><mi>u</mi><mo>(</mo><mi>i</mi><mo>)</mo><mo>=</mo><mi>v</mi><mo>(</mo><mi>i</mi><mo>)</mo><mo>}</mo><mo>|</mo><mo>≥</mo><mi>t</mi></math></span> for all <span><math><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>A</mi></math></span>. In this paper, we consider the diversity and other intersecting theorems for weak compositions.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167703","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}

{"title":"Bounding the intersection number c2 of a distance-regular graph with classical parameters (D,b,α,β) in terms of b","authors":"","doi":"10.1016/j.disc.2024.114239","DOIUrl":"10.1016/j.disc.2024.114239","url":null,"abstract":"<div><p>Let Γ be a distance-regular graph with classical parameters <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span> and <span><math><mi>b</mi><mo>≥</mo><mn>1</mn></math></span>. It is known that Γ is <em>Q</em>-polynomial with respect to <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>b</mi></mrow></mfrac><mo>−</mo><mn>1</mn></math></span> is the second largest eigenvalue of Γ. And it was shown that for a distance-regular graph Γ with classical parameters <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span>, <span><math><mi>D</mi><mo>≥</mo><mn>5</mn></math></span> and <span><math><mi>b</mi><mo>≥</mo><mn>1</mn></math></span>, if <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is large enough compared to <em>b</em> and Γ is thin, then the intersection number <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of Γ is bounded above by a function of <em>b</em>. In this paper, we obtain a similar result without the assumption that the graph Γ is thin.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167702","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}

{"title":"A digraph version of the Friendship Theorem","authors":"","doi":"10.1016/j.disc.2024.114238","DOIUrl":"10.1016/j.disc.2024.114238","url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003698/pdfft?md5=368b7d4c1379f8549152a904b901804b&pid=1-s2.0-S0012365X24003698-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167394","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}

{"title":"Extremal graphs for the odd prism","authors":"","doi":"10.1016/j.disc.2024.114249","DOIUrl":"10.1016/j.disc.2024.114249","url":null,"abstract":"<div><p>The Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> of a graph <em>H</em> is the maximum number of edges in an <em>n</em>-vertex graph which does not contain <em>H</em> 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 <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>□</mo></mrow></msubsup></math></span>, which is defined as the Cartesian product of an odd cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> and an edge <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Applying a deep theorem of Simonovits and a stability result of Yuan (2022) <span><span>[55]</span></span>, we shall determine the exact value of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mo>□</mo></mrow></msubsup><mo>)</mo></math></span> for every <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span> and sufficiently large <em>n</em>, and we also characterize the extremal graphs. Moreover, in the case of <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>, motivated by a recent result of Xiao et al. (2022) <span><span>[49]</span></span>, we will determine the exact value of <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow><mrow><mo>□</mo></mrow></msubsup><mo>)</mo></math></span> for every <em>n</em> instead of for sufficiently large <em>n</em>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003807/pdfft?md5=e2bc8fb4249126377f15948ed27aebbf&pid=1-s2.0-S0012365X24003807-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167396","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}

{"title":"The Jacobian of a graph and graph automorphisms","authors":"","doi":"10.1016/j.disc.2024.114259","DOIUrl":"10.1016/j.disc.2024.114259","url":null,"abstract":"<div><p>In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph <em>X</em> in the group of symmetries of the Jacobian of <em>X</em>. As a consequence we show that if a 3-edge-connected graph <em>X</em> admits a nonabelian semiregular group of automorphisms, then the Jacobian of <em>X</em> 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 <em>X</em> is well-understood – it is equal to the number of spanning trees of <em>X</em> – the combinatorial interpretation of the rank of Jacobian of a graph is unknown. Our paper presents a contribution in this direction.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142167704","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}