Discrete Mathematics最新文献

{"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}

{"title":"Triangular faces of the order and chain polytope of a maximal ranked poset","authors":"Aki Mori","doi":"10.1016/j.disc.2025.114480","DOIUrl":"10.1016/j.disc.2025.114480","url":null,"abstract":"<div><div>Let <span><math><mi>O</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> and <span><math><mi>C</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> denote the order polytope and chain polytope, respectively, associated with a finite poset <em>P</em>. We prove the following result: if <em>P</em> is a maximal ranked poset, then the number of triangular 2-faces of <span><math><mi>O</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span> is less than or equal to that of <span><math><mi>C</mi><mo>(</mo><mi>P</mi><mo>)</mo></math></span>, with equality holding if and only if <em>P</em> does not contain an <em>X</em>-poset as a subposet.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114480"},"PeriodicalIF":0.7,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143620425","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":"Claw-free cubic graphs are (1,1,2,2)-colorable","authors":"Boštjan Brešar ,&nbsp;Kirsti Kuenzel ,&nbsp;Douglas F. Rall","doi":"10.1016/j.disc.2025.114477","DOIUrl":"10.1016/j.disc.2025.114477","url":null,"abstract":"<div><div>A <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-coloring of a graph is a partition of its vertex set into four sets two of which are independent and the other two are 2-packings. In this paper, we prove that every claw-free cubic graph admits a <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-coloring. This implies that the conjecture from Brešar et al. (2017) <span><span>[5]</span></span> that the packing chromatic number of subdivisions of subcubic graphs is at most 5 is true in the case of claw-free cubic graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114477"},"PeriodicalIF":0.7,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143620424","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":"Sets of vertices with extremal energy","authors":"Neal Bushaw,&nbsp;Brent Cody,&nbsp;Chris Leffler","doi":"10.1016/j.disc.2025.114466","DOIUrl":"10.1016/j.disc.2025.114466","url":null,"abstract":"<div><div>We define various notions of energy of a set of vertices in a graph, which generalize two of the most widely studied graphical indices: the Wiener index and the Harary index. We provide a new proof of a result due to Douthett and Krantz, which says that for cycles, the sets of vertices which have minimal energy among all sets of the same size are precisely the <em>maximally even sets</em>, as defined in Clough and Douthett's work on music theory. Generalizing a theorem of Clough and Douthett, we prove that a finite, simple, connected graph is distance degree regular if and only if whenever a set of vertices has minimal energy, its complement also has minimal energy. We also provide several characterizations of sets of vertices in finite paths and cycles for which the sum of all pairwise distances between vertices in the set is maximal among all sets of the same size.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114466"},"PeriodicalIF":0.7,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562108","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":"N-ary groups of panmagic permutations from the Post coset theorem","authors":"Sergiy Koshkin ,&nbsp;Jaeho Lee","doi":"10.1016/j.disc.2025.114467","DOIUrl":"10.1016/j.disc.2025.114467","url":null,"abstract":"<div><div>Panmagic permutations are permutations whose matrices are panmagic squares, better known as maximal configurations of non-attacking queens on a toroidal chessboard. Some of them, affine panmagic permutations, can be conveniently described by linear formulas of modular arithmetic, and we show that their sets are a generalization of groups with <em>N</em>-ary multiplication instead of binary one. With the help of the Post coset theorem, we identify panmagic <em>N</em>-ary groups as cosets of the dihedral subgroup and its extensions in the group of all affine permutations. We also investigate decomposition of panmagic permutations into disjoint cycles and find many connections with classical topics of number theory and combinatorics: square-free numbers, <span><math><mn>4</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span> primes, quadratic residues, cycle indices from Polya counting, and linear congruential generators.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114467"},"PeriodicalIF":0.7,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563799","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 Galois LCD codes and LCPs of codes over mixed alphabets","authors":"Leijo Jose,&nbsp;Anuradha Sharma","doi":"10.1016/j.disc.2025.114465","DOIUrl":"10.1016/j.disc.2025.114465","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;R&lt;/span&gt; be a finite commutative chain ring with the maximal ideal &lt;span&gt;&lt;math&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of nilpotency index &lt;span&gt;&lt;math&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, and let &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; for some positive integer &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we study and characterize Galois &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-LCD codes of an arbitrary block-length. We show that each weakly-free &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-linear code is monomially equivalent to a Galois &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-LCD code when &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, while it is monomially equivalent to a Euclidean &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-LCD code when &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. We also obtain enumeration formulae for all Euclidean and Hermitian &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;-LCD codes of an arbitrary block-length. With the help of these enumeration formulae, we classify all Euclidean &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-LCD codes and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;9&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-LCD codes of block-lengths &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&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;, &lt;span&gt;&lt;math&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;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&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;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;3&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; and &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and all Hermitian &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-LCD codes of block-lengths &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&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;, &lt;span&gt;&lt;math&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;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114465"},"PeriodicalIF":0.7,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143563865","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}