{"title":"On Schur rings over infinite groups III","authors":"Nicholas Bastian, Andrew Misseldine","doi":"10.1007/s10801-023-01273-z","DOIUrl":"https://doi.org/10.1007/s10801-023-01273-z","url":null,"abstract":"","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134947169","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":"Edge ideals of Erdős–Rényi random graphs: linear resolution, unmixedness and regularity","authors":"Banerjee, Arindam, Yogeshwaran, D.","doi":"10.1007/s10801-023-01264-0","DOIUrl":"https://doi.org/10.1007/s10801-023-01264-0","url":null,"abstract":"We study the homological algebra of edge ideals of Erdős–Rényi random graphs. These random graphs are generated by deleting edges of a complete graph on n vertices independently of each other with probability $$1-p$$ . We focus on some aspects of these random edge ideals—linear resolution, unmixedness and algebraic invariants like the Castelnuovo–Mumford regularity, projective dimension and depth. We first show a double phase transition for existence of linear presentation and resolution and determine the critical windows as well. As a consequence, we obtain that except for a very specific choice of parameters (i.e., $$n,p:= p(n)$$ ), with high probability, a random edge ideal has linear presentation if and only if it has linear resolution. This shows certain conjectures hold true for large random graphs with high probability even though the conjectures were shown to fail for determinstic graphs. Next, we study asymptotic behaviour of some algebraic invariants—the Castelnuovo–Mumford regularity, projective dimension and depth—of such random edge ideals in the sparse regime (i.e., $$p = frac{lambda }{n}, lambda in (0,infty )$$ ). These invariants are studied using local weak convergence (or Benjamini-Schramm convergence) and relating them to invariants on Galton–Watson trees. We also show that when $$p rightarrow 0$$ or $$p rightarrow 1$$ fast enough, then with high probability the edge ideals are unmixed and for most other choices of p, these ideals are not unmixed with high probability. This is further progress towards the conjecture that random monomial ideals are unlikely to have Cohen–Macaulay property (De Loera et al. in Proc Am Math Soc 147(8):3239–3257, 2019; J Algebra 519:440–473, 2019) in the setting when the number of variables goes to infinity but the degree is fixed.","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"423-425 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134946889","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 groups with chordal power graph, including a classification in the case of finite simple groups","authors":"Jendrik Brachter, Eda Kaja","doi":"10.1007/s10801-023-01262-2","DOIUrl":"https://doi.org/10.1007/s10801-023-01262-2","url":null,"abstract":"Abstract We prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classified by (Electron J Combin 28(3):14, 2021). Here we classify the finite simple groups with chordal power graph, relative to typical number theoretic conditions. We do so by devising several sufficient conditions for the existence and non-existence of long cycles in power graphs of finite groups. We examine other natural group classes, including special linear, symmetric, generalized dihedral and quaternion groups, and we characterize direct products with chordal power graph. The classification problem is thereby reduced to directly indecomposable groups, and we further obtain a list of possible socles. Lastly, we give a general bound on the length of an induced path in chordal power graphs, providing another potential road to advance the classification beyond simple groups.","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135592076","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 popularity gap","authors":"Vsevolod F. Lev, Ilya D. Shkredov","doi":"10.1007/s10801-023-01268-w","DOIUrl":"https://doi.org/10.1007/s10801-023-01268-w","url":null,"abstract":"","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135552990","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":"Hexavalent edge-transitive graphs of order $$3p^2$$","authors":"Song-Tao Guo, Li Wang","doi":"10.1007/s10801-023-01265-z","DOIUrl":"https://doi.org/10.1007/s10801-023-01265-z","url":null,"abstract":"","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135592549","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":"Quantum state transfer between twins in weighted graphs","authors":"Steve Kirkland, Hermie Monterde, Sarah Plosker","doi":"10.1007/s10801-023-01261-3","DOIUrl":"https://doi.org/10.1007/s10801-023-01261-3","url":null,"abstract":"","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135688967","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":"Cutoff phenomenon for the warp-transpose top with random shuffle","authors":"Subhajit Ghosh","doi":"10.1007/s10801-023-01271-1","DOIUrl":"https://doi.org/10.1007/s10801-023-01271-1","url":null,"abstract":"","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135131717","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":"Characterisation of all integral circulant graphs with multiplicative divisor sets and few eigenvalues","authors":"J. W. Sander, T. Sander","doi":"10.1007/s10801-023-01259-x","DOIUrl":"https://doi.org/10.1007/s10801-023-01259-x","url":null,"abstract":"Abstract We present a method which in principal allows to characterise all integral circulant graphs with multiplicative divisor set having a spectrum, i.e. the set of distinct eigenvalues, of any given size. We shall exemplify the method for spectra of up to four eigenvalues, also reproving some known results for three eigenvalues along the way. In particular we show that given any integral circulant graph of arbitrary order n with multiplicative divisor set and precisely four distinct eigenvalues, n necessarily is either a prime power or the product of two prime powers with explicitly given simply structured divisor set and set of eigenvalues in both cases.","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136015935","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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