Discrete Mathematics最新文献

{"title":"Sufficient conditions for closed-trailable in digraphs","authors":"Changchang Dong ,&nbsp;Hong Yang ,&nbsp;Jixiang Meng ,&nbsp;Juan Liu","doi":"10.1016/j.disc.2025.114796","DOIUrl":"10.1016/j.disc.2025.114796","url":null,"abstract":"<div><div>A digraph <em>D</em> with a subset <em>S</em> of <span><math><mi>V</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span> is called <strong><em>S</em>-strong</strong> if for every pair of distinct vertices <em>u</em> and <em>v</em> of <em>S</em>, there is a <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span>-dipath and a <span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>)</mo></math></span>-dipath in <em>D</em>. We define a digraph <em>D</em> with a subset <em>S</em> of <span><math><mi>V</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span> to be <strong><em>S</em>-locally closed-trailable</strong> if <span><math><mi>D</mi><mo>〈</mo><mi>S</mi><mo>〉</mo></math></span> is a semicomplete digraph or there exist two nonadjacent vertices <span><math><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>S</mi></math></span> such that <em>D</em> contains a closed ditrail through the vertices <em>u</em> and <em>v</em>; and define a subset <span><math><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span> to be <strong>closed-trailable</strong> if <em>D</em> contains a closed ditrail through all the vertices of <em>S</em>. In this paper, we prove that for a digraph <em>D</em> with <em>n</em> vertices and a subset <em>S</em> of <span><math><mi>V</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span>, if <em>D</em> is <em>S</em>-strong and if <span><math><mi>d</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>+</mo><mi>d</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>≥</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>3</mn></math></span> for any two nonadjacent vertices <em>u</em> and <em>v</em> of <em>S</em>, then <em>S</em> is closed-trailable. This result generalizes the theorem of Bang-Jensen et al. <span><span>[3]</span></span> on supereulerianity. Moreover, we show that for a digraph <em>D</em> and a subset <em>S</em> of <span><math><mi>V</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span>, if <em>D</em> is <em>S</em>-locally closed-trailable and if <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>D</mi><mo>〈</mo><mi>S</mi><mo>〉</mo><mo>)</mo><mo>≥</mo><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>D</mi><mo>〈</mo><mi>S</mi><mo>〉</mo><mo>)</mo><mo>&gt;</mo><mn>0</mn></math></span>, where <span><math><msup><mrow><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>D</mi><mo>〈</mo><mi>S</mi><mo>〉</mo><mo>)</mo></math></span> is the minimum semi-degree of <span><math><mi>D</mi><mo>〈</mo><mi>S</mi><mo>〉</mo></math></span> and <span><math><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>D</mi><mo>〈</mo><mi>S</mi><mo>〉</mo><mo>)</mo></math></span> is the matching number of <span><math><mi>D</mi><mo>〈</mo><mi>S</mi><mo>〉</mo></math></span>, then <em>S</em> is closed-trailable. This result generalizes the theorem of Algefari et al. <span><span>[1]</span></span> on supereulerianity.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114796"},"PeriodicalIF":0.7,"publicationDate":"2025-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145107434","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":"Proper conflict-free degree-choosability of outerplanar graphs","authors":"Masaki Kashima ,&nbsp;Riste Škrekovski ,&nbsp;Rongxing Xu","doi":"10.1016/j.disc.2025.114800","DOIUrl":"10.1016/j.disc.2025.114800","url":null,"abstract":"<div><div>A proper coloring <em>ϕ</em> of <em>G</em> is called a proper conflict-free coloring of <em>G</em> if for every non-isolated vertex <em>v</em> of <em>G</em>, there is a color <em>c</em> such that <span><math><mo>|</mo><msup><mrow><mi>ϕ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>c</mi><mo>)</mo><mo>∩</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>=</mo><mn>1</mn></math></span>. As an analogy to degree-choosability of graphs, the authors recently, in a previous paper, introduced the notion of proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mi>k</mi><mo>)</mo></math></span>-choosability of graphs. For a non-negative integer <em>k</em>, a graph <em>G</em> is proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mi>k</mi><mo>)</mo></math></span>-choosable if for any list assignment <em>L</em> of <em>G</em> with <span><math><mo>|</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>|</mo><mo>≥</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>v</mi><mo>)</mo><mo>+</mo><mi>k</mi></math></span> for every vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <em>G</em> admits a proper conflict-free coloring <em>ϕ</em> such that <span><math><mi>ϕ</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>∈</mo><mi>L</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span> for every vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this paper, we show that every connected outerplanar graph other than the 5-cycle is proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mn>2</mn><mo>)</mo></math></span>-choosable. This bound is tight in the sense that there are infinitely many connected outerplanar graphs that are not proper conflict-free <span><math><mo>(</mo><mrow><mi>degree</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-choosable. We conclude the paper with two questions for further work.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114800"},"PeriodicalIF":0.7,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145107497","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":"Spectral conditions for spherical 2-distance sets","authors":"Iliyas Noman,&nbsp;Yuan Yao","doi":"10.1016/j.disc.2025.114795","DOIUrl":"10.1016/j.disc.2025.114795","url":null,"abstract":"<div><div>A set of points <em>S</em> in <em>d</em>-dimensional Euclidean space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> is called a <em>2-distance set</em> if the set of pairwise distances between the points has cardinality two. The 2-distance set is called <em>spherical</em> if its points lie on the unit sphere in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We characterize the spherical 2-distance sets using the spectrum of the adjacency matrix of an associated graph and the spectrum of the projection of the adjacency matrix onto the orthogonal complement of the all-ones vector. We also determine the lowest dimensional space in which a given spherical 2-distance set could be represented using the graph spectrum.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114795"},"PeriodicalIF":0.7,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145107496","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":"Counting cospectral graphs obtained via switching","authors":"Aida Abiad ,&nbsp;Nils van de Berg ,&nbsp;Robin Simoens","doi":"10.1016/j.disc.2025.114775","DOIUrl":"10.1016/j.disc.2025.114775","url":null,"abstract":"<div><div>Switching is an operation on a graph that does not change the spectrum of the adjacency matrix, thus producing cospectral graphs. An important activity in the field of spectral graph theory is the characterization of graphs by their spectrum. Thus switching provides a tool for disproving the existence of such a characterization.</div><div>This paper presents a general framework for counting the number of graphs that have a non-isomorphic cospectral graph through a switching method, expanding on the work by Haemers and Spence [European Journal of Combinatorics, 2004]. Our framework is based on a different counting approach, which allows it to be used for all known switching methods for the adjacency matrix. From this, we derive asymptotic results, which we complement with computer enumeration results for graphs up to 10 vertices.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114775"},"PeriodicalIF":0.7,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050242","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":"Three edge-disjoint plane spanning paths in a point set","authors":"Philipp Kindermann ,&nbsp;Jan Kratochvíl ,&nbsp;Giuseppe Liotta ,&nbsp;Pavel Valtr","doi":"10.1016/j.disc.2025.114780","DOIUrl":"10.1016/j.disc.2025.114780","url":null,"abstract":"<div><div>We consider the following problem: Given a set <em>S</em> of <em>n</em> distinct points in the plane, how many edge-disjoint plane straight-line spanning paths can be drawn on <em>S</em>? Each spanning path must be crossing-free, but edges from different paths are allowed to intersect at arbitrary points. It is known that if the points of <em>S</em> are in convex position, then <span><math><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></math></span> such paths always exist. However, for general point sets, the best known construction yields only two edge-disjoint plane spanning paths.</div><div>In this paper, we prove that for any set <em>S</em> of at least ten points in general position (i.e., no three points are collinear), it is always possible to draw at least three edge-disjoint plane straight-line spanning paths. Our proof relies on a structural result about halving lines in point sets and builds on the known two-path construction, which we also strengthen: we show that for any set <em>S</em> of at least six points, and for any two specified points on the boundary of the convex hull of <em>S</em>, there exist two edge-disjoint plane spanning paths that start at those prescribed points.</div><div>Finally, we complement our positive results with a lower bound: for every <span><math><mi>n</mi><mo>≥</mo><mn>6</mn></math></span>, there exists a set of <em>n</em> points for which no more than <span><math><mo>⌈</mo><mi>n</mi><mo>/</mo><mn>3</mn><mo>⌉</mo></math></span> edge-disjoint plane spanning paths are possible.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114780"},"PeriodicalIF":0.7,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050240","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":"Characterization of trees with Laplacian eigenvalue multiplicity one less than the number of pendant vertices","authors":"Dein Wong ,&nbsp;Wenhao Zhen,&nbsp;Songnian Xu","doi":"10.1016/j.disc.2025.114799","DOIUrl":"10.1016/j.disc.2025.114799","url":null,"abstract":"<div><div>Let <em>G</em> be a connected graph with vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and edge set <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. The Laplacian matrix of <em>G</em> is defined as <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the diagonal matrix with diagonal entries the degrees of the vertices of <em>G</em> and <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the adjacency matrix of <em>G</em>. The multiplicity of an eigenvalue <em>λ</em> of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is denoted by <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>λ</mi><mo>)</mo></math></span>. In 1990, Grone et al. (1990) <span><span>[10]</span></span> proved that <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>(</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for an arbitrary eigenvalue <em>λ</em> of <span><math><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span>, where <em>T</em> is a tree with at least two vertices. The authors of the above paper indicated that if <em>T</em> is a star <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> and <span><math><mi>λ</mi><mo>=</mo><mn>1</mn></math></span>, then the equality occurs. A natural problem is to characterize the trees <em>T</em> and the eigenvalues <em>λ</em> for which the aforementioned inequality holds as an equality. Till now, this problem is unsolved. In the present paper, we give a complete solution for the above problem.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 3","pages":"Article 114799"},"PeriodicalIF":0.7,"publicationDate":"2025-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050243","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 note on mixed cages of girth 5","authors":"Gabriela Araujo-Pardo ,&nbsp;Mirabel Mendoza-Cadena","doi":"10.1016/j.disc.2025.114773","DOIUrl":"10.1016/j.disc.2025.114773","url":null,"abstract":"<div><div>A <em>mixed regular graph</em> is a graph where every vertex has <em>z</em> incoming arcs, <em>z</em> outgoing arcs, and <em>r</em> edges; furthermore, if it has girth <em>g</em>, we say that the graph is a <span><math><mo>[</mo><mi>z</mi><mo>,</mo><mi>r</mi><mo>;</mo><mi>g</mi><mo>]</mo></math></span><em>-mixed graph</em>. A <span><math><mo>[</mo><mi>z</mi><mo>,</mo><mi>r</mi><mo>;</mo><mi>g</mi><mo>]</mo></math></span><em>-mixed cage</em> is a <span><math><mo>[</mo><mi>z</mi><mo>,</mo><mi>r</mi><mo>;</mo><mi>g</mi><mo>]</mo></math></span>-mixed graph with the smallest possible order. In this note, we give a family of <span><math><mo>[</mo><mi>z</mi><mo>,</mo><mi>q</mi><mo>;</mo><mn>5</mn><mo>]</mo></math></span>-mixed graphs for <span><math><mi>q</mi><mo>≥</mo><mn>7</mn></math></span> power of prime and <span><math><mi>q</mi><mo>−</mo><mn>1</mn><mo>≤</mo><mn>4</mn><mi>z</mi><mo>+</mo><mi>R</mi></math></span> with <span><math><mi>z</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>R</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>5</mn><mo>}</mo></math></span>. This provides better upper bounds on the order of mixed cages known until this moment.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114773"},"PeriodicalIF":0.7,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048611","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":"Dense maximal asymptotic nonbases of order two","authors":"Min Tang,&nbsp;Shi-Qiang Chen","doi":"10.1016/j.disc.2025.114781","DOIUrl":"10.1016/j.disc.2025.114781","url":null,"abstract":"<div><div>Let <em>A</em> be a set of nonnegative integers. The set <em>A</em> is called an asymptotic basis of order 2 if every sufficiently large integer can be written as the sum of two elements of <em>A</em>. Otherwise, <em>A</em> is called an asymptotic nonbasis of order 2. An asymptotic nonbasis <em>A</em> of order 2 is maximal if <span><math><mi>A</mi><mo>∪</mo><mo>{</mo><mi>b</mi><mo>}</mo></math></span> is an asymptotic basis of order 2 for every nonnegative integer <span><math><mi>b</mi><mo>∉</mo><mi>A</mi></math></span>. In this paper, we show that there exists a maximal asymptotic nonbasis <em>A</em> which contains consecutive integers and satisfies <span><math><mi>n</mi><mo>/</mo><mn>2</mn><mo>≤</mo><mi>A</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>≤</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn></math></span> for any nonnegative integer <em>n</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114781"},"PeriodicalIF":0.7,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048612","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":"Lattice paths enumerations weighted by ascent lengths","authors":"Jun Yan","doi":"10.1016/j.disc.2025.114783","DOIUrl":"10.1016/j.disc.2025.114783","url":null,"abstract":"<div><div>Recent work of the author connected several parking function enumeration problems to enumerations of Catalan paths with respect to certain weight functions that are expressed in terms of the ascent lengths. Motivated by this, we generalise and solve analogous weighted enumeration problems for a large family of lattice paths and weight functions, and discuss their connections with other enumeration problems and OEIS entries.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114783"},"PeriodicalIF":0.7,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048613","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 spectral generalized Alon-Frankl theorem","authors":"Zilong Yan ,&nbsp;Bingqing Yang,&nbsp;Yuejian Peng","doi":"10.1016/j.disc.2025.114785","DOIUrl":"10.1016/j.disc.2025.114785","url":null,"abstract":"<div><div>A recent result of Alon-Frankl determines the Turán number ex<span><math><mo>(</mo><mi>n</mi><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span>, i.e., the maximum number of edges in an <em>n</em>-vertex graph with clique number no more than <em>k</em> and matching number no more than <em>s</em>. Ma-Hou determined the generalized Turán number ex<span><math><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span>, i.e., the maximum number of copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> in an <em>n</em>-vertex graph with clique number no more than <em>k</em> and matching number no more than <em>s</em>. In the last two decades, the spectral Turán problems have also attracted considerable attention, but there are few studies on spectral versions of results on generalized Turán numbers. Our purpose in this paper is to give a spectral version of the generalized Alon-Frankl theorem via the hypergraph tensor.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114785"},"PeriodicalIF":0.7,"publicationDate":"2025-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048614","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}