{"title":"On the intersection spectrum of $${text {PSL}}_2(q)$$","authors":"Angelot Behajaina, Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra","doi":"10.1007/s10801-024-01356-5","DOIUrl":"https://doi.org/10.1007/s10801-024-01356-5","url":null,"abstract":"<p>Given a group <i>G</i> and a subgroup <span>(H le G)</span>, a set <span>(mathcal {F}subset G)</span> is called <i>H</i><i>-intersecting</i> if for any <span>(g,g' in mathcal {F})</span>, there exists <span>(xH in G/H)</span> such that <span>(gxH=g'xH)</span>. The <i>intersection density</i> of the action of <i>G</i> on <i>G</i>/<i>H</i> by (left) multiplication is the rational number <span>(rho (G,H))</span>, equal to the maximum ratio <span>(frac{|mathcal {F}|}{|H|})</span>, where <span>(mathcal {F} subset G)</span> runs through all <i>H</i>-intersecting sets of <i>G</i>. The <i>intersection spectrum</i> of the group <i>G</i> is then defined to be the set </p><span>$$begin{aligned} sigma (G) := left{ rho (G,H) : Hle G right} . end{aligned}$$</span><p>It was shown by Bardestani and Mallahi-Karai (J Algebraic Combin, 42(1):111–128, 2015) that if <span>(sigma (G) = {1})</span>, then <i>G</i> is necessarily solvable. The natural question that arises is, therefore, which rational numbers larger than 1 belong to <span>(sigma (G))</span>, whenever <i>G</i> is non-solvable. In this paper, we study the intersection spectrum of the linear group <span>({text {PSL}}_2(q))</span>. It is shown that <span>(2 in sigma left( {text {PSL}}_2(q)right) )</span>, for any prime power <span>(qequiv 3 pmod 4)</span>. Moreover, when <span>(qequiv 1 pmod 4)</span>, it is proved that <span>(rho ({text {PSL}}_2(q),H)=1)</span>, for any odd index subgroup <i>H</i> (containing <span>({mathbb {F}}_q)</span>) of the Borel subgroup (isomorphic to <span>({mathbb {F}}_qrtimes {mathbb {Z}}_{frac{q-1}{2}})</span>) consisting of all upper triangular matrices.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"52 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142264049","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":"Finite 4-geodesic-transitive graphs with bounded girth","authors":"Wei Jin, Li Tan","doi":"10.1007/s10801-024-01358-3","DOIUrl":"https://doi.org/10.1007/s10801-024-01358-3","url":null,"abstract":"<p>Praeger and the first author in Jin and Praeger (J Combin Theory Ser A 178:105349, 2021) asked the following problem: classify <i>s</i>-geodesic-transitive graphs of girth <span>(2s-1)</span> or <span>(2s-2)</span>, where <span>(s=4,5,6,7,8)</span>. In this paper, we study the <span>(s=4)</span> case, that is, study the family of finite (<i>G</i>, 4)-geodesic-transitive graphs of girth 6 or 7 for some group <i>G</i> of automorphisms. A reduction result on this family of graphs is first given. Let <i>N</i> be a normal subgroup of <i>G</i> which has at least 3 orbits on the vertex set. We show that such a graph <span>(Gamma )</span> is a cover of its quotient <span>(Gamma _N)</span> modulo the <i>N</i>-orbits and either <span>(Gamma _N)</span> is (<i>G</i>/<i>N</i>, <i>s</i>)-geodesic-transitive where <span>(s=min {4,textrm{diam}(Gamma _N)}ge 3)</span>, or <span>(Gamma _N)</span> is a (<i>G</i>/<i>N</i>, 2)-arc-transitive strongly regular graph. Next, using the classification of 2-arc-transitive strongly regular graphs, we determine all the (<i>G</i>, 4)-geodesic-transitive covers <span>(Gamma )</span> when <span>(Gamma _N)</span> is strongly regular.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"32 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203133","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":"Level and pseudo-Gorenstein path polyominoes","authors":"Giancarlo Rinaldo, Francesco Romeo, Rajib Sarkar","doi":"10.1007/s10801-024-01354-7","DOIUrl":"https://doi.org/10.1007/s10801-024-01354-7","url":null,"abstract":"<p>We classify path polyominoes which are level and pseudo-Gorenstein. Moreover, we compute all level and pseudo-Gorenstein simple thin polyominoes with rank less than or equal to 10. We also compute the regularity of the pseudo-Gorenstein simple thin polyominoes in relation to their rank.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"32 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203136","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":"Bipartite determinantal ideals and concurrent vertex maps","authors":"Li Li","doi":"10.1007/s10801-024-01351-w","DOIUrl":"https://doi.org/10.1007/s10801-024-01351-w","url":null,"abstract":"<p>Bipartite determinantal ideals are introduced in Illian and Li (Gröbner basis for the double determinantal ideals, http://arxiv.org/abs/2305.01724) as a vast generalization of the classical determinantal ideals intensively studied in commutative algebra, algebraic geometry, representation theory, and combinatorics. We introduce a combinatorial model called concurrent vertex maps to describe the Stanley–Reisner complex of the initial ideal of any bipartite determinantal ideal, and study properties and applications of this model including vertex decomposability, shelling orders, formulas of the Hilbert series, and <i>h</i>-polynomials.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"95 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203135","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}
Gregory Lupton, Oleg Musin, Nicholas A. Scoville, P. Christopher Staecker, Jonathan Treviño-Marroquín
{"title":"A second homotopy group for digital images","authors":"Gregory Lupton, Oleg Musin, Nicholas A. Scoville, P. Christopher Staecker, Jonathan Treviño-Marroquín","doi":"10.1007/s10801-024-01352-9","DOIUrl":"https://doi.org/10.1007/s10801-024-01352-9","url":null,"abstract":"<p>We define a second (higher) homotopy group for digital images. Namely, we construct a functor from digital images to abelian groups, which closely resembles the ordinary second homotopy group from algebraic topology. We illustrate that our approach can be effective by computing this (digital) second homotopy group for a digital 2-sphere.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"15 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142203134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A unified combinatorial view beyond some spectral properties","authors":"Xiaofeng Gu, Muhuo Liu","doi":"10.1007/s10801-024-01353-8","DOIUrl":"https://doi.org/10.1007/s10801-024-01353-8","url":null,"abstract":"<p>Let <span>(beta >0)</span>. Motivated by the notion of jumbled graphs introduced by Thomason, the expander mixing lemma and Haemers’s vertex separation inequality, we say that a graph <i>G</i> with <i>n</i> vertices is a weakly <span>((n,beta ))</span>-graph if <span>(frac{|X| |Y|}{(n-|X|)(n-|Y|)} le beta ^2)</span> holds for every pair of disjoint proper subsets <i>X</i>, <i>Y</i> of <i>V</i>(<i>G</i>) with no edge between <i>X</i> and <i>Y</i>. It is an <span>((n,beta ))</span>-graph if in addition <i>X</i> and <i>Y</i> are not necessarily disjoint. Using graph eigenvalues, we show that every graph can be an <span>((n,beta ))</span>-graph and/or a weakly <span>((n,beta ))</span>-graph for some specific value <span>(beta )</span>. For instances, the expander mixing lemma implies that a <i>d</i>-regular graph on <i>n</i> vertices with the second largest absolute eigenvalue at most <span>(lambda )</span> is an <span>((n,lambda /d))</span>-graph, and Haemers’s vertex separation inequality implies that every graph is a weakly <span>((n,beta ))</span>-graph with <span>(beta ge frac{mu _n - mu _2}{mu _n + mu _2})</span>, where <span>(mu _i)</span> denotes the <i>i</i>-th smallest Laplacian eigenvalue. This motivates us to study <span>((n,beta ))</span>-graph and weakly <span>((n,beta ))</span>-graph in general. Our main results include the following. (i) For any weakly <span>((n,beta ))</span>-graph <i>G</i>, the matching number <span>(alpha '(G)ge min left{ frac{1-beta }{1+beta },, frac{1}{2}right} cdot (n-1))</span>. If in addition <i>G</i> is a (<i>U</i>, <i>W</i>)-bipartite graph with <span>(|W|ge t|U|)</span> where <span>(tge 1)</span>, then <span>(alpha '(G)ge min {t(1-2beta ^2),1}cdot |U|)</span>. (ii) For any <span>((n,beta ))</span>-graph <i>G</i>, <span>(alpha '(G)ge min left{ frac{2-beta }{2(1+beta )},, frac{1}{2}right} cdot (n-1).)</span> If in addition <i>G</i> is a (<i>U</i>, <i>W</i>)-bipartite graph with <span>(|W|ge |U|)</span> and no isolated vertices, then <span>(alpha '(G)ge min {1/beta ^{2},1}cdot |U|)</span>. (iii) If <i>G</i> is a weakly <span>((n,beta ))</span>-graph for <span>(0<beta le 1/3)</span> or an <span>((n,beta ))</span>-graph for <span>(0<beta le 1/2)</span>, then <i>G</i> has a fractional perfect matching. In addition, <i>G</i> has a perfect matching when <i>n</i> is even and <i>G</i> is factor-critical when <i>n</i> is odd. (iv) For any connected <span>((n,beta ))</span>-graph <i>G</i>, the toughness <span>(t(G)ge frac{1-beta }{beta })</span>. For any connected weakly <span>((n,beta ))</span>-graph <i>G</i>, <span>(t(G)> frac{5(1-beta )}{11beta })</span> and if <i>n</i> is large enough, then <span>(t(G)>left( frac{1}{2}-varepsilon right) frac{1-beta }{beta })</span> for any <span>(varepsilon >0)</span>. The results imply many old and new results in spectral graph theory, including several new lower bounds on matching number, fractional matching number and toughness from eigenvalues. In particula","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226115","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":"The strongly robust simplicial complex of monomial curves","authors":"Dimitra Kosta, Apostolos Thoma, Marius Vladoiu","doi":"10.1007/s10801-024-01349-4","DOIUrl":"https://doi.org/10.1007/s10801-024-01349-4","url":null,"abstract":"<p>To every simple toric ideal <span>(I_T)</span> one can associate the strongly robust simplicial complex <span>(Delta _T)</span>, which determines the strongly robust property for all ideals that have <span>(I_T)</span> as their bouquet ideal. We show that for the simple toric ideals of monomial curves in <span>(mathbb {A}^{s})</span>, the strongly robust simplicial complex <span>(Delta _T)</span> is either <span>({emptyset })</span> or contains exactly one 0-dimensional face. In the case of monomial curves in <span>(mathbb {A}^{3})</span>, the strongly robust simplicial complex <span>(Delta _T)</span> contains one 0-dimensional face if and only if the toric ideal <span>(I_T)</span> is a complete intersection ideal with exactly two Betti degrees. Finally, we provide a construction to produce infinitely many strongly robust ideals with bouquet ideal the ideal of a monomial curve and show that they are all produced this way.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"78 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567649","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":"Tubes containing string modules in symmetric special multiserial algebras","authors":"Drew Damien Duffield","doi":"10.1007/s10801-024-01339-6","DOIUrl":"https://doi.org/10.1007/s10801-024-01339-6","url":null,"abstract":"<p>Symmetric special multiserial algebras are algebras that correspond to decorated hypergraphs with orientation, called Brauer configurations. In this paper, we use the combinatorics of Brauer configurations to understand the module category of symmetric special multiserial algebras via their Auslander–Reiten quiver. In particular, we provide methods for determining the existence and ranks of tubes in the stable Auslander–Reiten quiver of symmetric special multiserial algebras using only the information from the underlying Brauer configuration. Firstly, we define a combinatorial walk around the Brauer configuration, called a Green ‘hyperwalk’, which generalises the existing notion of a Green walk around a Brauer graph. Periodic Green hyperwalks are then shown to correspond to periodic projective resolutions of certain classes of string modules over the corresponding symmetric special multiserial algebra. Periodic Green hyperwalks thus determine certain classes of tubes in the stable Auslander–Reiten quiver, with the ranks of the tubes determined by the periods of the walks. Finally, we provide a description of additional rank two tubes in symmetric special multiserial algebras that do not arise from Green hyperwalks, but which nevertheless contain string modules at the mouth. This includes an explicit description of the space of extensions between string modules at the mouth of tubes of rank two.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"86 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567648","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":"Addition–deletion results for plus-one generated curves","authors":"Anca Măcinic, Piotr Pokora","doi":"10.1007/s10801-024-01350-x","DOIUrl":"https://doi.org/10.1007/s10801-024-01350-x","url":null,"abstract":"<p>In the recent paper A. Dimca proves that when one adds to or deletes a line from a free curve, the resulting curve is either free or plus-one generated. We prove the converse statements, we give an additional insight into the original deletion result, and we derive a characterization of free curves in terms of behavior to addition/deletion of lines. Incidentally we generalize a result on conic-line arrangements by H. Schenck and Ş. Tohăneanu that describes when the addition or the deletion of a projective line from a free curve results in a free curve. We catalogue the possible splitting types of the bundle of logarithmic vector fields associated to a plus-one generated curve.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"54 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567477","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":"The covariety of numerical semigroups with fixed Frobenius number","authors":"M. A. Moreno-Frías, J. C. Rosales","doi":"10.1007/s10801-024-01342-x","DOIUrl":"https://doi.org/10.1007/s10801-024-01342-x","url":null,"abstract":"<p>Denote by <span>({mathrm m}(S))</span> the multiplicity of a numerical semigroup <i>S</i>. A <i>covariety</i> is a nonempty family <span>(mathscr {C})</span> of numerical semigroups that fulfils the following conditions: there is the minimum of <span>(mathscr {C},)</span> the intersection of two elements of <span>(mathscr {C})</span> is again an element of <span>(mathscr {C})</span> and <span>(Sbackslash {{mathrm m}(S)}in mathscr {C})</span> for all <span>(Sin mathscr {C})</span> such that <span>(Sne min (mathscr {C}).)</span> In this work we describe an algorithmic procedure to compute all the elements of <span>(mathscr {C}.)</span> We prove that there exists the smallest element of <span>(mathscr {C})</span> containing a set of positive integers. We show that <span>(mathscr {A}(F)={Smid S hbox { is a numerical semigroup with Frobenius number }F})</span> is a covariety, and we particularize the previous results in this covariety. Finally, we will see that there is the smallest covariety containing a finite set of numerical semigroups.\u0000</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567476","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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