{"title":"The b-symbol weight hierarchies of two families of cyclic codes over Fq","authors":"Tonghui Zhang , Pinhui Ke , Zuling Chang","doi":"10.1016/j.disc.2025.114749","DOIUrl":"10.1016/j.disc.2025.114749","url":null,"abstract":"<div><div>Researchers are addressing the challenges posed by channels with overlapping symbol outputs by developing new encoding methods. Cassuto and Blaum introduced early encoding constructions in the symbol-pair metric, which was extended by Yaakobi, Bruck, and Siegel to the encoding of <em>b</em>-symbol metrics. This paper discusses the <em>b</em>-symbol weight hierarchies of two families of cyclic codes, one of which is completely determined. Two applications including the <em>b</em>-symbol MDS codes and the distance-optimal shortened codes are also provided.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114749"},"PeriodicalIF":0.7,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908688","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":"Fractional forcing number of graphs","authors":"Javad B. Ebrahimi , Babak Ghanbari","doi":"10.1016/j.disc.2025.114739","DOIUrl":"10.1016/j.disc.2025.114739","url":null,"abstract":"<div><div>The notion of forcing sets for perfect matchings was introduced by Harary, Klein, and Živković. The application of this problem in chemistry, as well as its interesting theoretical aspects, made this subject very active. In this work, we introduce the notion of forcing function of fractional perfect matchings, which is continuous analogous to forcing sets defined over the perfect matchings of graphs. We show that this object is a continuous and concave function extension of the integral forcing set. Then, we use our results in the continuous world to conclude new bounds and results in the discrete case of forcing sets, for the family of regular edge-transitive graphs. In particular, we derive new upper bounds for the maximum forcing number of hypercube graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114739"},"PeriodicalIF":0.7,"publicationDate":"2025-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144903082","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":"Isometric path complexity of graphs","authors":"Dibyayan Chakraborty , Jérémie Chalopin , Florent Foucaud , Yann Vaxès","doi":"10.1016/j.disc.2025.114743","DOIUrl":"10.1016/j.disc.2025.114743","url":null,"abstract":"<div><div>A set <em>S</em> of isometric paths of a graph <em>G</em> is “<em>v</em>-rooted”, where <em>v</em> is a vertex of <em>G</em>, if <em>v</em> is one of the endpoints of all the isometric paths in <em>S</em>. The <em>isometric path complexity</em> of a graph <em>G</em>, denoted by <span><math><mi>i</mi><mi>p</mi><mi>c</mi><mi>o</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math></span>, is the minimum integer <em>k</em> such that there exists a vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> satisfying the following property: the vertices of any single isometric path <em>P</em> of <em>G</em> can be covered by <em>k</em> many <em>v</em>-rooted isometric paths.</div><div>First, we provide an <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>m</mi><mo>)</mo></math></span>-time algorithm to compute the isometric path complexity of a graph with <em>n</em> vertices and <em>m</em> edges. Then we show that the isometric path complexity remains bounded for graphs in three seemingly unrelated graph classes, namely, <em>hyperbolic graphs</em>, <em>(theta, prism, pyramid)-free graphs</em>, and <em>outerstring graphs</em>. There is a direct algorithmic consequence of having small isometric path complexity. Specifically, we show that if the isometric path complexity of a graph <em>G</em> is bounded by a constant, then there exists a polynomial-time constant-factor approximation algorithm for <span>Isometric Path Cover</span>, whose objective is to cover all vertices of a graph with a minimum number of isometric paths. This applies to all the above graph classes.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114743"},"PeriodicalIF":0.7,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896206","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":"Cubical flips in quadrangulations of closed surfaces","authors":"Yusuke Suzuki","doi":"10.1016/j.disc.2025.114752","DOIUrl":"10.1016/j.disc.2025.114752","url":null,"abstract":"<div><div>In this paper, we discuss cubical flips defined for quadrangulations of closed surfaces, which are also known as cubical or bubble moves in combinatorial topology. In particular, we show that any two parity-congruent simple quadrangulations of a closed surface can be transformed into each other by a sequence of cubical flips. We then show that any two parity-congruent simple quadrangulations, excluding the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>k</mi></mrow></msub></math></span>, can be transformed into each other by a sequence of specific local transformations—namely, 33SPLs, 33CONs, Y-ROTs, and P-FLPs—while preserving simplicity throughout. Furthermore, we show that any two parity-congruent polyhedral quadrangulations can also be transformed into each other by cubical flips preserving polyhedrality. Lastly, we present some results on the minimality of the above local transformations under certain conditions.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114752"},"PeriodicalIF":0.7,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895701","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":"Ordering digraphs with given maximum outdegree by their Aα spectral radius","authors":"Zengzhao Xu , Weige Xi , Ligong Wang","doi":"10.1016/j.disc.2025.114744","DOIUrl":"10.1016/j.disc.2025.114744","url":null,"abstract":"<div><div>Let <em>G</em> be a strongly connected digraph with <em>n</em> vertices and <em>m</em> arcs. For any real <span><math><mi>α</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> matrix of a digraph <em>G</em> is defined as<span><span><span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>α</mi><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the adjacency matrix of <em>G</em> and <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the outdegrees diagonal matrix of <em>G</em>. The eigenvalue of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> with the largest modulus is called the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> spectral radius of <em>G</em>, denoted by <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this paper, we first obtain an upper bound on <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for <span><math><mi>α</mi><mo>∈</mo><mo>[</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. Employing this upper bound, we prove that for two strongly connected digraphs <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> with <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span> vertices and <em>m</em> arcs, and <span><math><mi>α</mi><mo>∈</mo><mo>[</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, if the maximum outdegree <span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>≥</mo><mn>2</mn><mi>α</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo><mo>(</mo><mi>m</mi><mo>−</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>2</mn><mi>α</mi></math></span> and <span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>></mo><msup><mrow><mi>Δ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, then <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114744"},"PeriodicalIF":0.7,"publicationDate":"2025-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144895700","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":"Murnaghan–Nakayama rules for symplectic, orthogonal and orthosymplectic Schur functions","authors":"Nishu Kumari , Anna Stokke","doi":"10.1016/j.disc.2025.114741","DOIUrl":"10.1016/j.disc.2025.114741","url":null,"abstract":"<div><div>We establish new Murnaghan–Nakayama rules for symplectic, orthogonal and orthosymplectic Schur functions. The classical Murnaghan–Nakayama rule expresses the product of a power sum symmetric function with a Schur function as a linear combination of Schur functions. Symplectic and orthogonal Schur functions correspond to characters of irreducible representations of symplectic and orthogonal groups. Orthosymplectic Schur functions arise as characters of orthosymplectic Lie superalgebras and are hybrids of symplectic and ordinary Schur functions. We derive explicit formulas for the product of the relevant power sum function with each of these functions, which can partly be described combinatorially using border strip manipulations. Our Murnaghan–Nakayama rules each include three distinct terms: a classical term corresponding to the addition of border strips to the relevant Young diagram, a term involving the removal of border strips, and a third term, which we describe both algebraically and combinatorially.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114741"},"PeriodicalIF":0.7,"publicationDate":"2025-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144896210","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 the relationship between (Af)α-spectral radii of graphs with starlike branch tree or bouquet branch graph and its linearly order","authors":"Xueliang Li, Ruiling Zheng","doi":"10.1016/j.disc.2025.114734","DOIUrl":"10.1016/j.disc.2025.114734","url":null,"abstract":"<div><div>For a graph <em>G</em> and a vertex <em>v</em> of <em>G</em>, let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></math></span> be the graph obtained from <em>G</em> by linking the paths on <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> vertices to the vertex <em>v</em> of <em>G</em>, respectively. We denote by <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> (or <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for short) the degree of the vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> in <em>G</em>. Let <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span> be a real symmetric function in <em>x</em> and <em>y</em>. The function-weighted adjacency matrix <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>f</mi></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> is a square matrix, where the <span><math><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></math></span>-entry is equal to <span><math><mi>f</mi><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></math></span> if the vertices <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> are adjacent and 0 otherwise, in which <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the degree of the vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. In <span><span>[22]</span></span>, Shan and Liu showed that the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>v</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow></math></span> will increase according to the shortlex ordering of <span><math><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</m","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114734"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886198","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":"Oriented colouring graphs of bounded degree and degeneracy","authors":"A. Clow, L. Stacho","doi":"10.1016/j.disc.2025.114746","DOIUrl":"10.1016/j.disc.2025.114746","url":null,"abstract":"<div><div>This paper considers upper bounds on the oriented chromatic number <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, of an oriented graph <em>G</em> in terms of its 2-dipath chromatic number <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, degeneracy <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and maximum degree <span><math><mi>Δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In particular, we show that for all graphs <em>G</em> with <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>k</mi></math></span> where <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>t</mi></math></span> where <span><math><mi>t</mi><mo>≥</mo><msub><mrow><mi>log</mi></mrow><mrow><mn>2</mn></mrow></msub><mo></mo><mo>(</mo><mi>k</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>33</mn><mo>/</mo><mn>10</mn><mo>(</mo><mi>k</mi><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi></mrow></msup><mo>)</mo></math></span>. This improves an upper bound of MacGillivray, Raspaud, and Swartz of the form <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msup><mo>−</mo><mn>1</mn></math></span> to a polynomial upper bound for many classes of graphs, in particular, those with bounded degeneracy. Additionally, we asymptotically improve bounds for the oriented chromatic number in terms of maximum degree and degeneracy. For instance, we show that <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mn>2</mn><mi>ln</mi><mo></mo><mn>2</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mn>2</mn></mrow><mrow><mi>Δ</mi></mrow></msup></math></span> for all graphs, and <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mn>2</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mi>Δ</mi><mi>d</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup></math></span> for graphs where degeneracy grows sublinearly in maximum degree. Here the asymptotics are in Δ. The former improves the asymptotics of a results by Kostochka, Sopena, and Zhu <span><span>[9]</span></span>, while the latter improves the asymptotics of a result by Aravind and Subramanian <span><span>[1]</span></span>. Both improvements are by a constant factor.</","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114746"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885662","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}
Dan Zeng , Yunge Xu , Lisha Li , Xianping Liu , Xiangyong Zeng
{"title":"The cycle structure of a class of permutation binomials","authors":"Dan Zeng , Yunge Xu , Lisha Li , Xianping Liu , Xiangyong Zeng","doi":"10.1016/j.disc.2025.114740","DOIUrl":"10.1016/j.disc.2025.114740","url":null,"abstract":"<div><div>In this paper, we study the cycle structure of permutation binomials of the form <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>θ</mi><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math></span> with 4-uniform DDT and 4-uniform BCT over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>. Note that the <em>k</em>-th composition of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> can be rewritten as the square root of a product of two polynomials <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math></span> and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mn>1</mn><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msup></math></span> for any <span><math><mi>x</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>, that is, <span><math><msup><mrow><mi>f</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span>. By further studying the periods of <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, we get the cycle structure of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114740"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885661","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":"Splitting fields of Schreier digraphs and t-Cayley hypergraphs","authors":"Yongjiang Wu , Weijun Liu , Lihua Feng , Xiaoqian Zhang","doi":"10.1016/j.disc.2025.114742","DOIUrl":"10.1016/j.disc.2025.114742","url":null,"abstract":"<div><div>In this paper, we determine the splitting fields and algebraic degrees of normal Schreier digraphs and certain <em>t</em>-Cayley hypergraphs. Additionally, we derive a decomposition formula for the spectra and distance spectra of <em>t</em>-Cayley hypergraphs over arbitrary finite groups, respectively. Furthermore, we determine the distance splitting fields and distance algebraic degrees of normal <em>t</em>-Cayley hypergraphs. These results generalize several previous work by Lu and Mönius on Cayley graphs and Sripaisan and Meemark on <em>t</em>-Cayley hypergraphs regarding splitting fields and algebraic degrees.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114742"},"PeriodicalIF":0.7,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144885728","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}
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