数学最新文献

{"title":"q-Analogs of divisible design graphs and Deza graphs","authors":"Dean Crnković ,&nbsp;Maarten De Boeck ,&nbsp;Francesco Pavese ,&nbsp;Andrea Švob","doi":"10.1016/j.jcta.2025.106047","DOIUrl":"10.1016/j.jcta.2025.106047","url":null,"abstract":"<div><div>Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. In this paper, we introduce the notion of <em>q</em>-analogs of divisible design graphs and show that all <em>q</em>-analogs of divisible design graphs come from spreads, and are actually <em>q</em>-analogs of strongly regular graphs.</div><div>Deza graphs were introduced by Erickson, Fernando, Haemers, Hardy and Hemmeter in 1999. In this paper, we introduce <em>q</em>-analogs of Deza graphs. Further, we determine possible parameters, give examples of <em>q</em>-analogs of Deza graphs and characterize all non-strongly regular <em>q</em>-analogs of Deza graphs with the smallest parameters.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106047"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantum integral Favard-type inequality","authors":"Jiao Yu","doi":"10.1016/j.amc.2025.129452","DOIUrl":"10.1016/j.amc.2025.129452","url":null,"abstract":"<div><div>In this paper, the classic Favard inequality in classical calculus is extended to quantum calculus, resulting in the quantum integral form of the Favard-type inequality. Furthermore, the quantum integral form under weighted conditions is considered. As <span><math><mi>q</mi><mo>→</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msup></math></span>, it degenerates into the classical Favard inequality.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"500 ","pages":"Article 129452"},"PeriodicalIF":3.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143760212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An atomic Coxeter presentation","authors":"Hankyung Ko","doi":"10.1016/j.aim.2025.110252","DOIUrl":"10.1016/j.aim.2025.110252","url":null,"abstract":"<div><div>We study parabolic double cosets in a Coxeter system by decomposing them into atom(ic coset)s, a generalization of simple reflections introduced in a joint work with Elias, Libedinsky, Patimo. We define and classify braid relations between compositions of atoms and prove a Matsumoto theorem. Together with a quadratic relation, our braid relations give a presentation of nilCoxeter algebroids similar to Demazure's presentation of nilCoxeter algebras. Our consideration of reduced compositions of atoms gives rise to a new combinatorial structure, which is equipped with a length function and a Bruhat order and is realized as Tits cone intersections in the sense of Iyama-Wemyss.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"470 ","pages":"Article 110252"},"PeriodicalIF":1.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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 maximum number of r-cliques in graphs free of complete r-partite subgraphs","authors":"József Balogh ,&nbsp;Suyun Jiang ,&nbsp;Haoran Luo","doi":"10.1016/j.disc.2025.114508","DOIUrl":"10.1016/j.disc.2025.114508","url":null,"abstract":"<div><div>We estimate the maximum possible number of cliques of size <em>r</em> in an <em>n</em>-vertex graph free of a fixed complete <em>r</em>-partite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></math></span>. By viewing every <em>r</em>-clique as a hyperedge, the upper bound on the Turán number of the complete <em>r</em>-partite hypergraphs gives the upper bound <span><math><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn><mo>/</mo><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mo>)</mo></mrow></math></span>. We improve this to <span><math><mi>o</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn><mo>/</mo><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mo>)</mo></mrow></math></span>. The main tool in our proof is the graph removal lemma. We also provide several lower bound constructions.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114508"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761179","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 bipartite graphs with the minimum number of spanning trees","authors":"Shicai Gong,&nbsp;Yue Xu,&nbsp;Peng Zou,&nbsp;Jiaxin Wang","doi":"10.1016/j.disc.2025.114514","DOIUrl":"10.1016/j.disc.2025.114514","url":null,"abstract":"<div><div>The collection of all (simple and connected) bipartite graphs with cyclomatic number <em>ω</em> is denoted by <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>. We use <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>;</mo><mi>b</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span> to denote the graph obtained from the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> by removing <span><math><mi>a</mi><mo>−</mo><mi>c</mi></math></span> edges that are all connected to the same vertex of degree <em>a</em>, here <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span> and <em>c</em> are integers with <span><math><mn>2</mn><mo>≤</mo><mi>c</mi><mo>&lt;</mo><mi>a</mi><mo>≤</mo><mi>b</mi></math></span>. The term <span><math><mi>S</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes the skeleton of the graph <em>G</em>, which is defined as the largest induced subgraph of <em>G</em> that contains no pendant vertices.</div><div>In this paper, we investigate the problem of characterizing the graphs within <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span> that possess the minimum number of spanning trees. We show that the skeleton of each graph with the minimum number of spanning trees in <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span> is either <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span>, where <em>a</em> and <em>b</em> are positive integers with <span><math><mn>2</mn><mo>≤</mo><mi>a</mi><mo>≤</mo><mi>b</mi></math></span> and <span><math><mo>(</mo><mi>a</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>ω</mi></math></span>, or <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>a</mi><mo>;</mo><mi>b</mi></mrow><mrow><mi>c</mi></mrow></msubsup></math></span>, where <span><math><mi>a</mi><mo>,</mo><mi>b</mi></math></span> and <em>c</em> are positive integers satisfying <span><math><mn>2</mn><mo>≤</mo><mi>c</mi><mo>&lt;</mo><mi>a</mi><mo>≤</mo><mi>b</mi></math></span> and <span><math><mi>c</mi><mo>−</mo><mn>1</mn><mo>+</mo><mo>(</mo><mi>a</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>ω</mi></math></span>. In addition, we establish some structural properties by the method of analysis to further reduce those candidate graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114514"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Topology and approximation of weak G-bundles in the supercritical dimensions","authors":"Swarnendu Sil","doi":"10.1016/j.aim.2025.110229","DOIUrl":"10.1016/j.aim.2025.110229","url":null,"abstract":"<div><div>For analyzing stationary Yang-Mills connections in higher dimensions, one has to work with Morrey-Sobolev bundles and connections. The transition maps for a Morrey-Sobolev principal <em>G</em>-bundles are not continuous and thus the usual notion of topology does not make sense. In this work, we develop the notion of a topological isomorphism class for a bundle-connection pair <span><math><mo>(</mo><mi>P</mi><mo>,</mo><mi>A</mi><mo>)</mo></math></span> and use these notions to derive several approximability results for bundles and connections in the Morrey-Sobolev setting. Our proofs follow a connection-oriented approach and also highlight the fact that in the low regularity regime, the regularity of the bundle and connection are intertwined. Our results parallel the theory of the topological degree and approximation results for manifold-valued VMO maps.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"470 ","pages":"Article 110229"},"PeriodicalIF":1.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The saturation number of C6","authors":"Yongxin Lan ,&nbsp;Yongtang Shi ,&nbsp;Yiqiao Wang ,&nbsp;Junxue Zhang","doi":"10.1016/j.disc.2025.114504","DOIUrl":"10.1016/j.disc.2025.114504","url":null,"abstract":"<div><div>A graph <em>G</em> is called <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-saturated if <em>G</em> is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-free but <span><math><mi>G</mi><mo>+</mo><mi>e</mi></math></span> is not for any <span><math><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>‾</mo></mover><mo>)</mo></math></span>. The saturation number of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, denoted <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span>, is the minimum number of edges in a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-saturated graph on <em>n</em> vertices. Finding the exact values of <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> has been one of the most intriguing open problems in extremal graph theory. In this paper, we study the saturation number of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span>. We prove that <span><math><mn>4</mn><mi>n</mi><mo>/</mo><mn>3</mn><mo>−</mo><mn>2</mn><mo>≤</mo><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>)</mo><mo>≤</mo><mo>(</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>3</mn></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>9</mn></math></span>, which significantly improves the existing lower and upper bounds for <span><math><mi>s</mi><mi>a</mi><mi>t</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114504"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761183","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":"Exceptional 2-to-1 rational functions","authors":"Zhiguo Ding ,&nbsp;Michael E. Zieve","doi":"10.1016/j.jcta.2025.106046","DOIUrl":"10.1016/j.jcta.2025.106046","url":null,"abstract":"<div><div>For each odd prime power <em>q</em>, we describe a class of rational functions <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the following unusual property: for every odd <em>j</em>, the function induced by <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msup></mrow></msub><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> is 2-to-1. We also show that, among all known rational functions <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> which are 2-to-1 on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msup></mrow></msub><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> for infinitely many <em>j</em>, our new functions are the only ones which cannot be written as compositions of rational functions of degree at most four, monomials, Dickson polynomials, and additive (linearized) polynomials.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106046"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parity statistics on restricted permutations and the Catalan–Schett polynomials","authors":"Zhicong Lin ,&nbsp;Jing Liu ,&nbsp;Sherry H.F. Yan","doi":"10.1016/j.jcta.2025.106049","DOIUrl":"10.1016/j.jcta.2025.106049","url":null,"abstract":"<div><div>Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over 321-avoiding permutations due to Barnabei, Bonetti and Silimbanian are obtained. In particular, an open problem of Kitaev and Zhang about non-overlapping ascents on 321-avoiding permutations is solved and several combinatorial interpretations for the Catalan–Schett polynomials are found. The stack-sortable permutations are at the heart of our approaches.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106049"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hyperbolic distortion and conformality at the boundary","authors":"Pavel Gumenyuk ,&nbsp;Maria Kourou ,&nbsp;Annika Moucha ,&nbsp;Oliver Roth","doi":"10.1016/j.aim.2025.110251","DOIUrl":"10.1016/j.aim.2025.110251","url":null,"abstract":"<div><div>We characterize two classical types of conformality of a holomorphic self-map of the unit disk at a boundary point — the existence of a finite angular derivative in the sense of Carathéodory and the weaker property of angle preservation — in terms of the non-tangential asymptotic behavior of the hyperbolic distortion of the map. We also provide an operator-theoretic characterization of the existence of a finite angular derivative based on Hilbert space methods. As an application we study the backward dynamics of discrete dynamical systems induced by holomorphic self-maps, and characterize the regularity of the associated pre-models in terms of a Blaschke-type condition involving the hyperbolic distortion along regular backward orbits.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"470 ","pages":"Article 110251"},"PeriodicalIF":1.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}