Pub Date : 2025-10-01Epub Date: 2025-04-03DOI: 10.1016/j.jcta.2025.106047
Dean Crnković , Maarten De Boeck , Francesco Pavese , Andrea Švob
Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. In this paper, we introduce the notion of q-analogs of divisible design graphs and show that all q-analogs of divisible design graphs come from spreads, and are actually q-analogs of strongly regular graphs.
Deza graphs were introduced by Erickson, Fernando, Haemers, Hardy and Hemmeter in 1999. In this paper, we introduce q-analogs of Deza graphs. Further, we determine possible parameters, give examples of q-analogs of Deza graphs and characterize all non-strongly regular q-analogs of Deza graphs with the smallest parameters.
{"title":"q-Analogs of divisible design graphs and Deza graphs","authors":"Dean Crnković , Maarten De Boeck , Francesco Pavese , 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":"2025-10-01","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}
Pub Date : 2025-10-01Epub Date: 2025-04-03DOI: 10.1016/j.jcta.2025.106046
Zhiguo Ding , Michael E. Zieve
For each odd prime power q, we describe a class of rational functions with the following unusual property: for every odd j, the function induced by on is 2-to-1. We also show that, among all known rational functions which are 2-to-1 on for infinitely many j, 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.
{"title":"Exceptional 2-to-1 rational functions","authors":"Zhiguo Ding , 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":"2025-10-01","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}
Pub Date : 2025-10-01Epub Date: 2025-04-03DOI: 10.1016/j.jcta.2025.106049
Zhicong Lin , Jing Liu , Sherry H.F. Yan
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.
{"title":"Parity statistics on restricted permutations and the Catalan–Schett polynomials","authors":"Zhicong Lin , Jing Liu , 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":"2025-10-01","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}
Pub Date : 2025-10-01Epub Date: 2025-03-13DOI: 10.1016/j.jcta.2025.106045
Edwin R. van Dam , Jack H. Koolen , Yanzhen Xiong
An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses. We show that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic. As a side result, we show that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it is amorphic.
{"title":"Characterizations of amorphic schemes and fusions of pairs","authors":"Edwin R. van Dam , Jack H. Koolen , Yanzhen Xiong","doi":"10.1016/j.jcta.2025.106045","DOIUrl":"10.1016/j.jcta.2025.106045","url":null,"abstract":"<div><div>An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the set of relations, where a pair forms an edge if it fuses. We show that if the fusing-relations graph is connected but not a path, then the association scheme is amorphic. As a side result, we show that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it is amorphic.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106045"},"PeriodicalIF":0.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143611456","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}
Pub Date : 2025-10-01Epub Date: 2025-04-30DOI: 10.1016/j.jcta.2025.106061
Bence Csajbók , Zoltán Lóránt Nagy
A subset S of an abelian group G is called 3-AP free if it does not contain a three term arithmetic progression. Moreover, S is called complete 3-AP free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a 3-AP free set, which is necessarily complete. In this paper we are interested in the minimum size of complete 3-AP free sets. We define and study saturation w.r.t. 3-APs and present constructions of small complete 3-AP free sets and 3-AP saturating sets for several families of vector spaces and cyclic groups.
{"title":"Complete 3-term arithmetic progression free sets of small size in vector spaces and other abelian groups","authors":"Bence Csajbók , Zoltán Lóránt Nagy","doi":"10.1016/j.jcta.2025.106061","DOIUrl":"10.1016/j.jcta.2025.106061","url":null,"abstract":"<div><div>A subset <em>S</em> of an abelian group <em>G</em> is called 3-AP free if it does not contain a three term arithmetic progression. Moreover, <em>S</em> is called complete 3-AP free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a 3-AP free set, which is necessarily complete. In this paper we are interested in the minimum size of complete 3-AP free sets. We define and study saturation w.r.t. 3-APs and present constructions of small complete 3-AP free sets and 3-AP saturating sets for several families of vector spaces and cyclic groups.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106061"},"PeriodicalIF":0.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886651","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}
Let Γ be a Cayley graph on a finite group G, and let be the normalizer of (the right regular representation of G) in the full automorphism group of Γ. We say that Γ is a normal Cayley graph on G if , and that Γ is a normal edge-transitive Cayley graph on G if acts transitively on the edge set of Γ. In 1999, Praeger proved that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency 3 is normal. As an extension of this, in this paper, we prove that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency p is normal for each prime p. This, however, is not true for composite valency. We give a method to construct connected normal edge-transitive but non-normal Cayley graphs of certain groups, and using this, we prove that if G is either for an odd prime , or for , then there exists a connected normal edge-transitive but non-normal 8-valent Cayley graph of G.
{"title":"Normal edge-transitive Cayley graphs on non-abelian simple groups","authors":"Xing Zhang, Yan-Quan Feng, Fu-Gang Yin, Jin-Xin Zhou","doi":"10.1016/j.jcta.2025.106050","DOIUrl":"10.1016/j.jcta.2025.106050","url":null,"abstract":"<div><div>Let <em>Γ</em> be a Cayley graph on a finite group <em>G</em>, and let <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> be the normalizer of <span><math><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (the right regular representation of <em>G</em>) in the full automorphism group <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> of <em>Γ</em>. We say that <em>Γ</em> is a normal Cayley graph on <em>G</em> if <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>=</mo><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span>, and that <em>Γ</em> is a normal edge-transitive Cayley graph on <em>G</em> if <span><math><msub><mrow><mi>N</mi></mrow><mrow><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow></msub><mo>(</mo><mi>R</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> acts transitively on the edge set of <em>Γ</em>. In 1999, Praeger proved that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency 3 is normal. As an extension of this, in this paper, we prove that every connected normal edge-transitive Cayley graph on a finite non-abelian simple group of valency <em>p</em> is normal for each prime <em>p</em>. This, however, is not true for composite valency. We give a method to construct connected normal edge-transitive but non-normal Cayley graphs of certain groups, and using this, we prove that if <em>G</em> is either <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> for an odd prime <span><math><mi>q</mi><mo>≥</mo><mn>5</mn></math></span>, or <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>, then there exists a connected normal edge-transitive but non-normal 8-valent Cayley graph of <em>G</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106050"},"PeriodicalIF":0.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739311","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}
Pub Date : 2025-10-01Epub Date: 2025-03-18DOI: 10.1016/j.jcta.2025.106035
Heng Huat Chan , Song Heng Chan
In this article, we discuss finite versions of Euler's pentagonal number identity, the Rogers-Ramanujan identities and present new proofs of the finite versions of the Andrews-Gordon identity and the Bressoud identity. We also investigate the finite version of Garvan's generalizations of Dyson's rank and discover a new one-variable extension of the Andrews-Gordon identity.
{"title":"Finite versions of the Andrews–Gordon identity and Bressoud's identity","authors":"Heng Huat Chan , Song Heng Chan","doi":"10.1016/j.jcta.2025.106035","DOIUrl":"10.1016/j.jcta.2025.106035","url":null,"abstract":"<div><div>In this article, we discuss finite versions of Euler's pentagonal number identity, the Rogers-Ramanujan identities and present new proofs of the finite versions of the Andrews-Gordon identity and the Bressoud identity. We also investigate the finite version of Garvan's generalizations of Dyson's rank and discover a new one-variable extension of the Andrews-Gordon identity.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106035"},"PeriodicalIF":0.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643721","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}
Pub Date : 2025-10-01Epub Date: 2025-04-30DOI: 10.1016/j.jcta.2025.106060
Aleksei L. Perezhogin , Vladimir N. Potapov , Anna A. Taranenko , Sergey Yu. Vladimirov
A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to 1. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if d is even, then the permanent of a d-dimensional polystochastic matrix of order 4 is positive, and for odd d, we give a complete characterization of d-dimensional polystochastic matrices with zero permanent.
{"title":"Characterization of polystochastic matrices of order 4 with zero permanent","authors":"Aleksei L. Perezhogin , Vladimir N. Potapov , Anna A. Taranenko , Sergey Yu. Vladimirov","doi":"10.1016/j.jcta.2025.106060","DOIUrl":"10.1016/j.jcta.2025.106060","url":null,"abstract":"<div><div>A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to 1. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if <em>d</em> is even, then the permanent of a <em>d</em>-dimensional polystochastic matrix of order 4 is positive, and for odd <em>d</em>, we give a complete characterization of <em>d</em>-dimensional polystochastic matrices with zero permanent.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106060"},"PeriodicalIF":0.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886650","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}
Pub Date : 2025-10-01Epub Date: 2025-05-02DOI: 10.1016/j.jcta.2025.106063
Jianrong Zhao , Chenxu Wang , Yu Fu
<div><div>Let <span><math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> be a gcd-closed set (i.e. <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>S</mi></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math></span>). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets <em>S</em> with <span><math><mo>|</mo><mi>S</mi><mo>|</mo><mo>≥</mo><mn>4</mn></math></span> such that the GCD matrix (<em>S</em>) divides the LCM matrix <span><math><mo>[</mo><mi>S</mi><mo>]</mo></math></span> in the ring <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. For <span><math><mi>x</mi><mo>∈</mo><mi>S</mi></math></span>, let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>S</mi><mo>:</mo><mi>z</mi><mo><</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>|</mo><mi>x</mi><mtext> and </mtext><mo>(</mo><mi>z</mi><mo>|</mo><mi>y</mi><mo>|</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>)</mo><mo>⇒</mo><mi>y</mi><mo>∈</mo><mo>{</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>}</mo><mo>}</mo></math></span>. In 2009, Feng, Hong and Zhao answered this problem in the context where <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≤</mo><mn>2</mn></math></span>. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set <em>S</em> with <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>=</mo><mn>3</mn></math></span> such that <span><math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math></span>. Meanwhile, they raised a conjecture on the necessary and sufficient condition such that <span><math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math></span> holds for the remaining case <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≥</mo><mn>4</mn></math></span>. In this paper, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely.</d
设S={x1，…，xn}是一个gcd闭集（即(xi,xj)∈S，对于所有1≤i，j≤n）。2002年，Hong提出了表征所有GCD -闭集S的|S|≥4使得GCD矩阵(S)能除环Mn(Z)中的LCM矩阵[S]的可分性问题。x∈年代,让GS (x): = {z∈年代:z< x, z | x和y z | | x, y∈(S)⇒y∈{z、x}}。2009年，Feng， Hong和Zhao在maxx∈S∈{|GS(x)|}≤2的情况下回答了这个问题。Zhao、Chen和Hong在2022年得到了maxx∈S∈S (|GS(x)|}=3的gcd-闭集S上的一个充要条件，使得(S)|[S]。同时，他们提出了一个关于(S)|[S]对剩余情况maxx∈S∈{|GS(x)|}≥4成立的充分必要条件的猜想。本文从一个全新的角度证实了赵-陈-洪猜想，从而彻底解决了洪的开放问题。
{"title":"Studying the divisibility of power LCM matrices by power GCD matrices on gcd-closed sets","authors":"Jianrong Zhao , Chenxu Wang , Yu Fu","doi":"10.1016/j.jcta.2025.106063","DOIUrl":"10.1016/j.jcta.2025.106063","url":null,"abstract":"<div><div>Let <span><math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> be a gcd-closed set (i.e. <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>S</mi></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math></span>). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets <em>S</em> with <span><math><mo>|</mo><mi>S</mi><mo>|</mo><mo>≥</mo><mn>4</mn></math></span> such that the GCD matrix (<em>S</em>) divides the LCM matrix <span><math><mo>[</mo><mi>S</mi><mo>]</mo></math></span> in the ring <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. For <span><math><mi>x</mi><mo>∈</mo><mi>S</mi></math></span>, let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>S</mi><mo>:</mo><mi>z</mi><mo><</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>|</mo><mi>x</mi><mtext> and </mtext><mo>(</mo><mi>z</mi><mo>|</mo><mi>y</mi><mo>|</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>)</mo><mo>⇒</mo><mi>y</mi><mo>∈</mo><mo>{</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>}</mo><mo>}</mo></math></span>. In 2009, Feng, Hong and Zhao answered this problem in the context where <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≤</mo><mn>2</mn></math></span>. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set <em>S</em> with <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>=</mo><mn>3</mn></math></span> such that <span><math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math></span>. Meanwhile, they raised a conjecture on the necessary and sufficient condition such that <span><math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math></span> holds for the remaining case <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≥</mo><mn>4</mn></math></span>. In this paper, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely.</d","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106063"},"PeriodicalIF":0.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895609","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}
Pub Date : 2025-08-08DOI: 10.1016/j.jcta.2025.106097
Yuxuan Li, Binzhou Xia, Sanming Zhou
A Cayley graph on the symmetric group <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. For <mml:math altimg="si2.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>n</mml:mi></mml:math>, let <mml:math altimg="si267.svg"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> be the set of <ce:italic>k</ce:italic>-cycles of <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> moving every point in <mml:math altimg="si4.svg"><mml:mo stretchy="false">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">}</mml:mo></mml:math>. Recently, Siemons and Zalesski (2022) <ce:cross-ref ref>[26]</ce:cross-ref> posed a conjecture which is equivalent to saying that for any <mml:math altimg="si5.svg"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>5</mml:mn></mml:math> and <mml:math altimg="si2.svg"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak="goodbreak" linebreakstyle="after"><</mml:mo><mml:mi>n</mml:mi></mml:math> the nonnormal Cayley graph <mml:math altimg="si6.svg"><mml:mrow><mml:mi mathvariant="normal">Cay</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> on <mml:math altimg="si1.svg"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> with connection set <mml:math altimg="si267.svg"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) <mml:math altimg="si7.svg"><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi
如果对称群Sn上的Cayley图的严格第二大特征值（即严格小于度的最大特征值）通过Sn的标准表示获得，则称其具有Aldous性质。对于1≤r<；k<n，设C（n,k;r）为Sn移动{1，…，r}中每一点的k个环的集合。最近，Siemons and Zalesski(2022)[26]提出了一个猜想，该猜想等价于对于任意n≥5且1≤r<；k<n，具有连接集C（n,k;r）的Sn上的非正态Cayley图Cay（Sn，C(n,k;r)）具有Aldous性质。通过求解这个猜想，我们证明了除(i) (n,k,r)=(6,5,1)或（ii） n为奇数，k =n−1，且1≤r<；n2外，所有图都具有Aldous性质。在此过程中，我们确定了Sn的所有不可约表示，这些表示可以实现Cay（Sn，C(n,n - 1;r)）的严格第二大特征值以及该图的最小特征值。
{"title":"The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups","authors":"Yuxuan Li, Binzhou Xia, Sanming Zhou","doi":"10.1016/j.jcta.2025.106097","DOIUrl":"https://doi.org/10.1016/j.jcta.2025.106097","url":null,"abstract":"A Cayley graph on the symmetric group <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>. For <mml:math altimg=\"si2.svg\"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>n</mml:mi></mml:math>, let <mml:math altimg=\"si267.svg\"><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> be the set of <ce:italic>k</ce:italic>-cycles of <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> moving every point in <mml:math altimg=\"si4.svg\"><mml:mo stretchy=\"false\">{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">}</mml:mo></mml:math>. Recently, Siemons and Zalesski (2022) <ce:cross-ref ref>[26]</ce:cross-ref> posed a conjecture which is equivalent to saying that for any <mml:math altimg=\"si5.svg\"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>5</mml:mn></mml:math> and <mml:math altimg=\"si2.svg\"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>r</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>k</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\"><</mml:mo><mml:mi>n</mml:mi></mml:math> the nonnormal Cayley graph <mml:math altimg=\"si6.svg\"><mml:mrow><mml:mi mathvariant=\"normal\">Cay</mml:mi></mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo><mml:mo stretchy=\"false\">)</mml:mo></mml:math> on <mml:math altimg=\"si1.svg\"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> with connection set <mml:math altimg=\"si267.svg\"><mml:mi>C</mml:mi><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>;</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:math> has the Aldous property. Solving this conjecture, we prove that all these graphs have the Aldous property except when (i) <mml:math altimg=\"si7.svg\"><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"22 1","pages":"106097"},"PeriodicalIF":1.1,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144900111","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}
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