Discrete Mathematics最新文献

{"title":"The king degree and the second out-degree of tournaments","authors":"Aya Alhussein ,&nbsp;Ayman El Zein","doi":"10.1016/j.disc.2025.114497","DOIUrl":"10.1016/j.disc.2025.114497","url":null,"abstract":"<div><div>In a digraph, the second out-degree of a vertex <em>x</em>, denoted by <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, is the number of vertices <em>y</em> such that <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span>, where <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is the length of the shortest <em>xy</em>-directed path, if it exists. It is obvious that the sum of the first out-degrees of the vertices in a digraph is nothing but the number of its arcs. Unlike the first out-degree, the summation of the second out-degrees of the vertices in a digraph is not constant with respect to the number of vertices and arcs. In this paper, we characterize, as a function of some integer <em>n</em>, the values that can be the summation of the second out-degrees of the vertices in a tournament of order <em>n</em>. Throughout the paper, we use the new concept of king degree in order to settle the problem. The king degree of a vertex <em>x</em> is the number of vertices that can be reached from <em>x</em> by a directed path of length at most 2. Several open problems are introduced in the last section of the paper.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114497"},"PeriodicalIF":0.7,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683710","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":"Combinatorics on words and generating Dirichlet series of automatic sequences","authors":"Jean-Paul Allouche ,&nbsp;Jeffrey Shallit ,&nbsp;Manon Stipulanti","doi":"10.1016/j.disc.2025.114487","DOIUrl":"10.1016/j.disc.2025.114487","url":null,"abstract":"<div><div>Generating series are crucial in enumerative combinatorics, analytic combinatorics, and combinatorics on words. Though it might seem at first view that generating Dirichlet series are less used in these fields than ordinary and exponential generating series, there are many notable papers where they play a fundamental role, as can be seen in particular in the work of Flajolet and several of his co-authors. In this paper, we study Dirichlet series of integers with missing digits or blocks of digits in some integer base <em>b</em>; i.e., where the summation ranges over the integers whose expansions form some language strictly included in the set of all words over the alphabet <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span> that do not begin with a 0. We show how to unify and extend results proved by Nathanson in 2021 and by Köhler and Spilker in 2009. En route, we encounter several sequences from Sloane's On-Line Encyclopedia of Integer Sequences, as well as some famous <em>b</em>-automatic sequences or <em>b</em>-regular sequences. We also consider a specific sequence that is not <em>b</em>-regular.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114487"},"PeriodicalIF":0.7,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143684325","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":"Sign-twisted generating functions of the odd length for Weyl groups of type D","authors":"Haihang Gu ,&nbsp;Houyi Yu","doi":"10.1016/j.disc.2025.114494","DOIUrl":"10.1016/j.disc.2025.114494","url":null,"abstract":"<div><div>The odd length in Weyl groups is a new statistic analogous to the classical Coxeter length, and features combinatorial and parity conditions. We establish explicit closed product formulas for the sign-twisted generating functions of the odd length for parabolic quotients of Weyl groups of type <em>D</em>. As a consequence, we verify three conjectures of Brenti and Carnevale on evaluating closed forms for these generating functions. We then give an equivalent condition for the sign-twisted generating functions to be expressible as products of cyclotomic polynomials, settling a conjecture of Stembridge.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114494"},"PeriodicalIF":0.7,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683711","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":"EA-cordial labeling of graphs and its implications for A-antimagic labeling of trees","authors":"Sylwia Cichacz","doi":"10.1016/j.disc.2025.114493","DOIUrl":"10.1016/j.disc.2025.114493","url":null,"abstract":"<div><div>If <em>A</em> is a finite Abelian group, then a labeling <span><math><mi>f</mi><mo>:</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mi>A</mi></math></span> of the edges of some graph <em>G</em> induces a vertex labeling on <em>G</em>; the vertex <em>u</em> receives the label <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>N</mi><mo>(</mo><mi>u</mi><mo>)</mo></mrow></msub><mi>f</mi><mo>(</mo><mi>u</mi><mi>v</mi><mo>)</mo></math></span>, where <span><math><mi>N</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span> is an open neighborhood of the vertex <em>u</em>. A graph <em>G</em> is <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span>-cordial if there is an edge-labeling such that (1) the edge label classes differ in size by at most one and (2) the induced vertex label classes differ in size by at most one. Such a labeling is called <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span>-cordial. In the literature, so far only <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span>-cordial labeling in cyclic groups has been studied.</div><div>Kaplan, Lev, and Roditty studied the corresponding problem. Namely, they introduced <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-antimagic labeling as a generalization of antimagic labeling <span><span>[11]</span></span>. Simply saying, for a tree of order <span><math><mo>|</mo><mi>A</mi><mo>|</mo></math></span> the <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-antimagic labeling is such <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span>-cordial labeling that the label 0 is prohibited on the edges.</div><div>In this paper, we give necessary and sufficient conditions for paths to be <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span>-cordial for any cyclic <em>A</em>. We also show that the conjecture for <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-antimagic labeling of trees posted in <span><span>[11]</span></span> is not true.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114493"},"PeriodicalIF":0.7,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683709","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":"Edge-regular graphs with fixed smallest eigenvalue with an application to Neumaier graphs","authors":"Qianqian Yang ,&nbsp;Jack H. Koolen","doi":"10.1016/j.disc.2025.114489","DOIUrl":"10.1016/j.disc.2025.114489","url":null,"abstract":"<div><div>In this paper, we will show that edge-regular graphs with fixed smallest eigenvalue and large valency are highly structured. As a consequence, we will prove that there are only finitely many strictly Neumaier graphs with smallest eigenvalue at least −3.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114489"},"PeriodicalIF":0.7,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685845","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":"Optimal locally repairable codes with multiple repair sets based on 2-regular packings","authors":"Jinghui Zhao,&nbsp;Yifei Li,&nbsp;Xiuling Shan","doi":"10.1016/j.disc.2025.114499","DOIUrl":"10.1016/j.disc.2025.114499","url":null,"abstract":"<div><div>Locally repairable codes can improve the repair efficiency in distributed storage system. In this paper, we consider the locally repairable codes with multiple disjoint repair sets and each repair set contains exactly one check symbol. We obtain optimal locally repairable codes with <span><math><mi>d</mi><mo>=</mo><mi>δ</mi><mo>=</mo><mn>3</mn></math></span> by constructing 2-regular packings. Firstly, we present several 2-regular packings by using two special configurations. Then we apply these 2-regular packings to construct optimal <span><math><msub><mrow><mo>[</mo><mi>k</mi><mo>+</mo><mo>⌈</mo><mfrac><mrow><mn>2</mn><mi>k</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>⌉</mo><mo>,</mo><mi>k</mi><mo>,</mo><mn>3</mn><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> systematic codes with information <span><math><msub><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mn>3</mn><mo>;</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>c</mi></mrow></msub></math></span>-locality.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114499"},"PeriodicalIF":0.7,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683708","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":"Bounds for the trace norm of Aα matrix of digraphs","authors":"Mushtaq A. Bhat,&nbsp;Peer Abdul Manan","doi":"10.1016/j.disc.2025.114491","DOIUrl":"10.1016/j.disc.2025.114491","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;D&lt;/em&gt; be a digraph of order &lt;em&gt;n&lt;/em&gt; with adjacency matrix &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. For &lt;span&gt;&lt;math&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, the &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; matrix of &lt;em&gt;D&lt;/em&gt; is defined as &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mtext&gt;diag&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is the diagonal matrix of vertex out degrees of &lt;em&gt;D&lt;/em&gt;. Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&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;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be the singular values of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Then the trace norm of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, which we call &lt;em&gt;α&lt;/em&gt; trace norm of &lt;em&gt;D&lt;/em&gt;, is defined as &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&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;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;σ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we find the singular values of some basic digraphs and characterize the digraphs &lt;em&gt;D&lt;/em&gt; with &lt;span&gt;&lt;math&gt;&lt;mtext&gt;Rank&lt;/mtext&gt;&lt;mspace&gt;&lt;/mspace&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;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. As an application of these results, we obtain a lower bound for the trace norm of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; matrix of digraphs and determine the extremal digraphs. In particular, we determine the oriented trees for which the trace norm of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114491"},"PeriodicalIF":0.7,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143684395","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":"Cyclic balanced sampling plans avoiding adjacent units with block size four","authors":"Zian Zhang,&nbsp;Yanxun Chang ,&nbsp;Tao Feng","doi":"10.1016/j.disc.2025.114490","DOIUrl":"10.1016/j.disc.2025.114490","url":null,"abstract":"<div><div>Balanced sampling plans avoiding adjacent units can be utilized for survey sampling when the units are arranged in one-dimensional ordering and the adjacent units in this ordering provide similar information. The existence of such a balanced sampling plan is equivalent to the existence of a <em>k</em>-clique decomposition of a special Cayley graph over a cyclic group. This paper completely determines the spectrum of cyclic balanced sampling plans avoiding adjacent units with maximum distance five and block size four.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114490"},"PeriodicalIF":0.7,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643767","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":"On regular graphs with a tree of diameter 3 as a star complement","authors":"Peter Rowlinson ,&nbsp;Zoran Stanić","doi":"10.1016/j.disc.2025.114488","DOIUrl":"10.1016/j.disc.2025.114488","url":null,"abstract":"<div><div>We investigate the regular graphs with a star complement <em>H</em> which is a tree of diameter 3. Thus <em>H</em> is a double star <span><math><mi>D</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, i.e. a tree with two vertices of degree <em>m</em> and <em>n</em> greater than 1, and all other vertices of degree 1. We determine all the regular graphs <em>G</em> that arise when either (a) <span><math><mi>μ</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></math></span> or (b) <span><math><mi>m</mi><mo>=</mo><mi>n</mi></math></span> and <em>μ</em> is an integer less than −1. It is also proved that for <span><math><mi>m</mi><mo>=</mo><mi>n</mi></math></span> and <span><math><mi>μ</mi><mo>≥</mo><mn>2</mn></math></span>, the degree of <em>G</em> must be <em>n</em>; moreover,<span><span><span><math><mi>n</mi><mo>≥</mo><mfrac><mrow><mi>μ</mi><mo>(</mo><mi>μ</mi><mo>(</mo><mn>2</mn><mi>μ</mi><mo>(</mo><mi>μ</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>−</mo><mn>3</mn><mo>)</mo><mo>+</mo><mn>3</mn><mo>)</mo><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>μ</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>,</mo></math></span></span></span> when <em>μ</em> is an integer.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114488"},"PeriodicalIF":0.7,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143631788","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":"Pentavalent 2-regular core-free Cayley graphs","authors":"Bo Ling,&nbsp;Zhi Ming Long","doi":"10.1016/j.disc.2025.114479","DOIUrl":"10.1016/j.disc.2025.114479","url":null,"abstract":"<div><div>A Cayley graph <span><math><mi>Γ</mi><mo>=</mo><mi>Cay</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> is said to be 2-regular core-free if <em>G</em> is core-free in some <span><math><mi>X</mi><mo>⩽</mo><mi>Aut</mi><mspace></mspace><mi>Γ</mi></math></span> and <span><math><mi>Aut</mi><mspace></mspace><mi>Γ</mi></math></span> acts regularly on the set of 2-arcs of <em>Γ</em>. In this paper, we classify the pentavalent 2-regular core-free Cayley graphs. As a byproduct, we provide another proof of one of the results by Du et al. (2017) <span><span>[6]</span></span> regarding pentavalent symmetric graphs over non-abelian simple groups. Namely, we prove that the pentavalent 2-regular Cayley graphs over non-abelian simple groups are normal. Furthermore, we construct a pentavalent core-free 2-transitive Cayley graph <span><math><mi>Cay</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> such that <span><math><mi>Aut</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> is transitive but not 2-transitive on <em>S</em>. This answers a question posed by Li in 2008.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114479"},"PeriodicalIF":0.7,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143637764","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}