Journal of Combinatorial Theory Series A最新文献

{"title":"Positivity and tails of pentagonal number series","authors":"Nian Hong Zhou","doi":"10.1016/j.jcta.2024.105933","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105933","url":null,"abstract":"<div><p>In this paper, we refine a result of Andrews and Merca on truncated pentagonal number series. Subsequently, we establish some positivity results involving Andrews–Gordon–Bressoud identities and <em>d</em>-regular partitions. In particular, we prove several conjectures of Merca and Krattenthaler–Merca–Radu on truncated pentagonal number series.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141541065","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":"On the difference of the enhanced power graph and the power graph of a finite group","authors":"Sucharita Biswas ,&nbsp;Peter J. Cameron ,&nbsp;Angsuman Das ,&nbsp;Hiranya Kishore Dey","doi":"10.1016/j.jcta.2024.105932","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105932","url":null,"abstract":"<div><p>The difference graph <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a finite group <em>G</em> is the difference of the enhanced power graph of <em>G</em> and the power graph of <em>G</em>, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of <span><math><mi>D</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> with respect to various properties of the underlying group <em>G</em>. We also find several connections between the difference graph of <em>G</em> and the Gruenberg-Kegel graph of <em>G</em>. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141439067","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":"Flag-transitive automorphism groups of 2-designs with λ ≥ (r,λ)2 are not product type","authors":"Huiling Li ,&nbsp;Zhilin Zhang ,&nbsp;Shenglin Zhou","doi":"10.1016/j.jcta.2024.105923","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105923","url":null,"abstract":"<div><p>In this note we show that a flag-transitive automorphism group <em>G</em> of a non-trivial 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design with <span><math><mi>λ</mi><mo>≥</mo><msup><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> is not of product action type. In conclusion, <em>G</em> is a primitive group of affine or almost simple type.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141429448","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":"Intersection density of imprimitive groups of degree pq","authors":"Angelot Behajaina ,&nbsp;Roghayeh Maleki ,&nbsp;Andriaherimanana Sarobidy Razafimahatratra","doi":"10.1016/j.jcta.2024.105922","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105922","url":null,"abstract":"<div><p>A subset <span><math><mi>F</mi></math></span> of a finite transitive group <span><math><mi>G</mi><mo>≤</mo><mi>Sym</mi><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> is <em>intersecting</em> if any two elements of <span><math><mi>F</mi></math></span> agree on an element of Ω. The <em>intersection density</em> of <em>G</em> is the number<span><span><span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo>⁡</mo><mrow><mo>{</mo><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow><mo>/</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub><mo>|</mo><mo>|</mo><mi>F</mi><mo>⊂</mo><mi>G</mi><mtext> is intersecting</mtext><mo>}</mo></mrow><mo>,</mo></math></span></span></span> where <span><math><mi>ω</mi><mo>∈</mo><mi>Ω</mi></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span> is the stabilizer of <em>ω</em> in <em>G</em>. It is known that if <span><math><mi>G</mi><mo>≤</mo><mi>Sym</mi><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> is an imprimitive group of degree a product of two odd primes <span><math><mi>p</mi><mo>&gt;</mo><mi>q</mi></math></span> admitting a block of size <em>p</em> or two complete block systems, whose blocks are of size <em>q</em>, then <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>.</p><p>In this paper, we analyze the intersection density of imprimitive groups of degree <em>pq</em> with a unique block system with blocks of size <em>q</em> based on the kernel of the induced action on blocks. For those whose kernels are non-trivial, it is proved that the intersection density is larger than 1 whenever there exists a cyclic code <em>C</em> with parameters <span><math><msub><mrow><mo>[</mo><mi>p</mi><mo>,</mo><mi>k</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> such that any codeword of <em>C</em> has weight at most <span><math><mi>p</mi><mo>−</mo><mn>1</mn></math></span>, and under some additional conditions on the cyclic code, it is a proper rational number. For those that are quasiprimitive, we reduce the cases to almost simple groups containing <span><math><mi>Alt</mi><mo>(</mo><mn>5</mn><mo>)</mo></math></span> or a projective special linear group. We give some examples where the latter has intersection density equal to 1, under some restrictions on <em>p</em> and <em>q</em>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S009731652400061X/pdfft?md5=4600ca58b59525e76de9f361f9c870b7&pid=1-s2.0-S009731652400061X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141308391","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":"Remarks on MacMahon's q-series","authors":"Ken Ono,&nbsp;Ajit Singh","doi":"10.1016/j.jcta.2024.105921","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105921","url":null,"abstract":"<div><p>In his important 1920 paper on partitions, MacMahon defined the partition generating functions<span><span><span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></munderover><mi>m</mi><mo>(</mo><mi>k</mi><mo>;</mo><mi>n</mi><mo>)</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>0</mn><mo>&lt;</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mo>⋯</mo><mo>&lt;</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munder><mfrac><mrow><msup><mrow><mi>q</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>k</mi></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>⋯</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo></math></span></span></span><span><span><span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></munderover><msub><mrow><mi>m</mi></mrow><mrow><mi>odd</mi></mrow></msub><mo>(</mo><mi>k</mi><mo>;</mo><mi>n</mi><mo>)</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mn>0</mn><mo>&lt;</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mo>⋯</mo><mo>&lt;</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></munder><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>2</mn><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><mn>2</mn><msub><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><mi>k</mi></mrow></msup></mrow><mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mn>1<","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141240350","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":"A multiparametric Murnaghan-Nakayama rule for Macdonald polynomials","authors":"Naihuan Jing ,&nbsp;Ning Liu","doi":"10.1016/j.jcta.2024.105920","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105920","url":null,"abstract":"<div><p>We introduce a new family of operators as multi-parameter deformation of the one-row Macdonald polynomials. The matrix coefficients of these operators acting on the space of symmetric functions with rational coefficients in two parameters <span><math><mi>q</mi><mo>,</mo><mi>t</mi></math></span> (denoted by <span><math><mi>Λ</mi><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>) are computed by assigning some values to skew Macdonald polynomials in <em>λ</em>-ring notation. The new rule is utilized to provide new iterative formulas and also recover various existing formulas in a unified manner. Specifically the following applications are discussed: (i) A <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-Murnaghan-Nakayama rule for Macdonald functions is given as a generalization of the <em>q</em>-Murnaghan-Nakayama rule; (ii) An iterative formula for the <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-Green polynomial is deduced; (iii) A simple proof of the Murnaghan-Nakayama rule for the Hecke algebra and the Hecke-Clifford algebra is offered; (iv) A combinatorial inversion of the Pieri rule for Hall-Littlewood functions is derived with the help of the vertex operator realization of the Hall-Littlewood functions; (v) Two iterative formulae for the <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-Kostka polynomials <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>λ</mi><mi>μ</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> are obtained from the dual version of our multiparametric Murnaghan-Nakayama rule, one of which yields an explicit formula for arbitrary <em>λ</em> and <em>μ</em> in terms of the generalized <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-binomial coefficient introduced independently by Lassalle and Okounkov.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141163388","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":"A characterisation of edge-affine 2-arc-transitive covers of K2n,2n","authors":"Daniel R. Hawtin ,&nbsp;Cheryl E. Praeger ,&nbsp;Jin-Xin Zhou","doi":"10.1016/j.jcta.2024.105919","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105919","url":null,"abstract":"<div><p>The family of finite 2-arc-transitive graphs of a given valency is closed under forming non-trivial <em>normal quotients</em>, and graphs in this family having no non-trivial normal quotient are called ‘basic’. To date, the vast majority of work in the literature has focused on classifying these ‘basic’ graphs. By contrast we give here a characterisation of the normal covers of the ‘basic’ 2-arc-transitive graphs <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>. The characterisation identified the special role of graphs associated with a subgroup of automorphisms called an <em>n-dimensional mixed dihedral group</em>. This is a group <em>H</em> with two subgroups <em>X</em> and <em>Y</em>, each elementary abelian of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>, such that <span><math><mi>X</mi><mo>∩</mo><mi>Y</mi><mo>=</mo><mn>1</mn></math></span>, <em>H</em> is generated by <span><math><mi>X</mi><mo>∪</mo><mi>Y</mi></math></span>, and <span><math><mi>H</mi><mo>/</mo><msup><mrow><mi>H</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≅</mo><mi>X</mi><mo>×</mo><mi>Y</mi></math></span>.</p><p>Our characterisation shows that each 2-arc-transitive normal cover of <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> is either itself a Cayley graph, or is the line graph of a Cayley graph of an <em>n</em>-dimensional mixed dihedral group. In the latter case, we show that the 2-arc-transitive group acting on the normal cover of <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> induces an <em>edge-affine</em> action on <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> (and we show that such actions are one of just four possible types of 2-arc-transitive actions on <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>). As a partial converse, we provide a graph theoretic characterisation of <em>n</em>-dimensional mixed dihedral groups, and finally, for each <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, we give an explicit construction of an <em>n</em>-dimensional mixed dihedral group which is a 2-group of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><m","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141163387","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":"Power-free complementary binary morphisms","authors":"Jeffrey Shallit ,&nbsp;Arseny Shur ,&nbsp;Stefan Zorcic","doi":"10.1016/j.jcta.2024.105910","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105910","url":null,"abstract":"<div><p>We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary morphisms. Such morphisms are defined over a 2-letter alphabet, and map the letters 0 and 1 to complementary words. We prove that every prefix of the famous Thue–Morse word <strong>t</strong> gives a complementary morphism that is <span><math><msup><mrow><mn>3</mn></mrow><mrow><mo>+</mo></mrow></msup></math></span>-free and hence <em>α</em>-free for every real number <span><math><mi>α</mi><mo>&gt;</mo><mn>3</mn></math></span>. We also describe, using a 4-state binary finite automaton, the lengths of all prefixes of <strong>t</strong> that give cubefree complementary morphisms. Next, we show that 3-free (cubefree) complementary morphisms of length <em>k</em> exist for all <span><math><mi>k</mi><mo>∉</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>6</mn><mo>}</mo></math></span>. Moreover, if <em>k</em> is not of the form <span><math><mn>3</mn><mo>⋅</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>, then the images of letters can be chosen to be factors of <strong>t</strong>. Finally, we observe that each cubefree complementary morphism is also <em>α</em>-free for some <span><math><mi>α</mi><mo>&lt;</mo><mn>3</mn></math></span>; in contrast, no binary morphism that maps each letter to a word of length 3 (resp., a word of length 6) is <em>α</em>-free for any <span><math><mi>α</mi><mo>&lt;</mo><mn>3</mn></math></span>.</p><p>In addition to more traditional techniques of combinatorics on words, we also rely on the Walnut theorem-prover. Its use and limitations are discussed.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141083216","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":"Spectra of power hypergraphs and signed graphs via parity-closed walks","authors":"Lixiang Chen ,&nbsp;Edwin R. van Dam ,&nbsp;Changjiang Bu","doi":"10.1016/j.jcta.2024.105909","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105909","url":null,"abstract":"<div><p>The <em>k</em>-power hypergraph <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> is the <em>k</em>-uniform hypergraph that is obtained by adding <span><math><mi>k</mi><mo>−</mo><mn>2</mn></math></span> new vertices to each edge of a graph <em>G</em>, for <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>. A parity-closed walk in <em>G</em> is a closed walk that uses each edge an even number of times. In an earlier paper, we determined the eigenvalues of the adjacency tensor of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> using the eigenvalues of signed subgraphs of <em>G</em>. Here, we express the entire spectrum (that is, we determine all multiplicities and the characteristic polynomial) of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> in terms of parity-closed walks of <em>G</em>. Moreover, we give an explicit expression for the multiplicity of the spectral radius of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span>. As a side result, we show that the number of parity-closed walks of given length is the corresponding spectral moment averaged over all signed graphs with underlying graph <em>G</em>. By extrapolating the characteristic polynomial of <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> to <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>, we introduce a pseudo-characteristic function which is shown to be the geometric mean of the characteristic polynomials of all signed graphs on <em>G</em>. This supplements a result by Godsil and Gutman that the arithmetic mean of the characteristic polynomials of all signed graphs on <em>G</em> equals the matching polynomial of <em>G</em>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141078251","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":"Maximum flag-rank distance codes","authors":"Gianira N. Alfarano ,&nbsp;Alessandro Neri ,&nbsp;Ferdinando Zullo","doi":"10.1016/j.jcta.2024.105908","DOIUrl":"https://doi.org/10.1016/j.jcta.2024.105908","url":null,"abstract":"<div><p>In this paper we extend the study of linear spaces of upper triangular matrices endowed with the flag-rank metric. Such metric spaces are isometric to certain spaces of degenerate flags and have been suggested as suitable framework for network coding. In this setting we provide a Singleton-like bound which relates the parameters of a flag-rank-metric code. This allows us to introduce the family of maximum flag-rank distance codes, that are flag-rank-metric codes meeting the Singleton-like bound with equality. Finally, we provide several constructions of maximum flag-rank distance codes.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000475/pdfft?md5=2c125aba1bc7cdaa56a76a0c7c4abe5d&pid=1-s2.0-S0097316524000475-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141078250","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}