{"title":"Realisation of groups as automorphism groups in permutational categories","authors":"G. Jones","doi":"10.26493/1855-3974.1840.6E0","DOIUrl":"https://doi.org/10.26493/1855-3974.1840.6E0","url":null,"abstract":"It is shown that in various categories, including many consisting of maps or hypermaps, oriented or unoriented, of a given hyperbolic type, or of coverings of a suitable topological space, every countable group A is isomorphic to the automorphism group of uncountably many non-isomorphic objects, infinitely many of them finite if A is finite. In particular, the latter applies to dessins d’enfants, regarded as finite oriented hypermaps.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90414156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aleksander Kelenc, Aoden Teo Masa Toshi, R. Škrekovski, I. Yero
{"title":"On metric dimensions of hypercubes","authors":"Aleksander Kelenc, Aoden Teo Masa Toshi, R. Škrekovski, I. Yero","doi":"10.26493/1855-3974.2568.55c","DOIUrl":"https://doi.org/10.26493/1855-3974.2568.55c","url":null,"abstract":"The metric (resp. edge metric or mixed metric) dimension of a graph $G$, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of $G$ by using a vector of distances to this set. In this note we show two unexpected results on hypercube graphs. First, we show that the metric and edge metric dimension of $Q_d$ differ by only one for every integer $d$. In particular, if $d$ is odd, then the metric and edge metric dimensions of $Q_d$ are equal. Second, we prove that the metric and mixed metric dimensions of the hypercube $Q_d$ are equal for every $d ge 3$. We conclude the paper by conjecturing that all these three types of metric dimensions of $Q_d$ are equal when $d$ is large enough.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85122371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two-distance transitive normal Cayley graphs","authors":"Junzhi Huang, Yan-Quan Feng, Jin-Xin Zhou","doi":"10.26493/1855-3974.2593.1B7","DOIUrl":"https://doi.org/10.26493/1855-3974.2593.1B7","url":null,"abstract":"In this paper, we construct an infinite family of normal Cayley graphs, which are 2 -distance-transitive but neither distance-transitive nor 2 -arc-transitive. This answers a question proposed by Chen, Jin and Li in 2019.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73466677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Oriented regular representations of out-valency two for finite simple groups","authors":"Gabriel Verret, Binzhou Xia","doi":"10.26493/1855-3974.2558.173","DOIUrl":"https://doi.org/10.26493/1855-3974.2558.173","url":null,"abstract":"In this paper, we show that every finite simple group of order at least 5 admits an oriented regular representation of out-valency 2.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74913214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On complete multipartite derangement graphs","authors":"A. S. Razafimahatratra","doi":"10.26493/1855-3974.2554.856","DOIUrl":"https://doi.org/10.26493/1855-3974.2554.856","url":null,"abstract":"Given a finite transitive permutation group $Gleq operatorname{Sym}(Omega)$, with $|Omega|geq 2$, the derangement graph $Gamma_G$ of $G$ is the Cayley graph $operatorname{Cay}(G,operatorname{Der}(G))$, where $operatorname{Der}(G)$ is the set of all derangements of $G$. Meagher et al. [On triangles in derangement graphs, J. Combin. Theory, Ser. A, 180:105390, 2021] recently proved that $operatorname{Sym}(2)$ acting on ${1,2}$ is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also showed that there exist transitive groups whose derangement graphs are complete multipartite. \u0000This paper gives two new families of transitive groups with complete multipartite derangement graphs. In addition, we construct an infinite family of transitive groups whose derangement graphs are multipartite but not complete.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91382743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On generalized Minkowski arrangements","authors":"M'at'e Kadlicsk'o, Z. L'angi","doi":"10.26493/1855-3974.2550.d96","DOIUrl":"https://doi.org/10.26493/1855-3974.2550.d96","url":null,"abstract":"The concept of a Minkowski arrangement was introduced by Fejes T'oth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes T'oth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order $mu$ for some $0<mu<1$ if no member $K$ of the family overlaps the homothetic copy of any other member $K'$ with ratio $mu$ and with the same center as $K'$. In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order $mu$ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes T'oth for Minkowski arrangements of circular disks, and a result of B\"or\"oczky and Szab'o about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87761883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some remarks on the square graph of the hypercube","authors":"S. Mirafzal","doi":"10.26493/1855-3974.2621.26f","DOIUrl":"https://doi.org/10.26493/1855-3974.2621.26f","url":null,"abstract":"Let $Gamma=(V,E)$ be a graph. The square graph $Gamma^2$ of the graph $Gamma$ is the graph with the vertex set $V(Gamma^2)=V$ in which two vertices are adjacent if and only if their distance in $Gamma$ is at most two. The square graph of the hypercube $Q_n$ has some interesting properties. For instance, it is highly symmetric and panconnected. In this paper, we investigate some algebraic properties of the graph ${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$ is distance-transitive. We show that the graph ${Q^2_n}$ is an imprimitive distance-transitive graph if and only if $n$ is an odd integer. Also, we determine the spectrum of the graph $Q_n^2$. Finally, we show that when $n>2$ is an even integer, then ${Q^2_n}$ is an automorphic graph, that is, $Q_n^2$ is a distance-transitive primitive graph which is not a complete or a line graph.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73761572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
L. K. Jørgensen, Guillermo Pineda-Villavicencio, J. Ugon
{"title":"Linkedness of Cartesian products of complete graphs","authors":"L. K. Jørgensen, Guillermo Pineda-Villavicencio, J. Ugon","doi":"10.26493/1855-3974.2577.25d","DOIUrl":"https://doi.org/10.26493/1855-3974.2577.25d","url":null,"abstract":"This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. \u0000We show that the Cartesian product $K^{d_{1}+1}times K^{d_{2}+1}$ of complete graphs $K^{d_{1}+1}$ and $K^{d_{2}+1}$ is $floor{(d_{1}+d_{2})/2}$-linked for $d_{1},d_{2}ge 2$, and this is best possible. \u0000%A polytope is said to be {it $k$-linked} if its graph is $k$-linked. \u0000This result is connected to graphs of simple polytopes. The Cartesian product $K^{d_{1}+1}times K^{d_{2}+1}$ is the graph of the Cartesian product $T(d_{1})times T(d_{2})$ of a $d_{1}$-dimensional simplex $T(d_{1})$ and a $d_{2}$-dimensional simplex $T(d_{2})$. And the polytope $T(d_{1})times T(d_{2})$ is a {it simple polytope}, a $(d_{1}+d_{2})$-dimensional polytope in which every vertex is incident to exactly $d_{1}+d_{2}$ edges. \u0000While not every $d$-polytope is $floor{d/2}$-linked, it may be conjectured that every simple $d$-polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86056326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Jonathan E. Leech: Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond","authors":"Jonathan Leech","doi":"10.26493/978-961-293-027-1","DOIUrl":"https://doi.org/10.26493/978-961-293-027-1","url":null,"abstract":"About the book: The extended study of non-commutative lattices was begun in 1949 by Ernst Pascual Jordan, a theoretical and mathematical physicist and co-worker of Max Born and Werner Karl Heisenberg. Jordan introduced noncommutative lattices as algebraic structures potentially suitable to encompass the logic of the quantum world. The modern theory of noncommutative lattices began forty years later with Jonathan Leech’s 1989 paper “Skew lattices in rings.” Recently, noncommutative generalizations of lattices and related structures have seen an upsurge in interest, with new ideas and applications emerging, from quasilattices to skew Heyting algebras. Much of this activity is derived in some way from the initiation of Jonathan Leech’s program of research in this area. The present book consists of seven chapters, mainly covering skew lattices, quasilattices and paralattices, skew lattices of idempotents in rings and skew Boolean algebras. As such, it is the first research monograph covering major results due to this renewed study of noncommutative lattices. It will serve as a valuable graduate textbook on the subject, as well as a handy reference to researchers of noncommutative algebras.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77056454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Point-primitive generalised hexagons and octagons and projective linear groups","authors":"S. Glasby, Emilio Pierro, C. Praeger","doi":"10.26493/1855-3974.2049.3DB","DOIUrl":"https://doi.org/10.26493/1855-3974.2049.3DB","url":null,"abstract":"We discuss recent progress on the problem of classifying point-primitive generalised polygons. In the case of generalised hexagons and generalised octagons, this has reduced the problem to primitive actions of almost simple groups of Lie type. To illustrate how the natural geometry of these groups may be used in this study, we show that if $mathcal{S}$ is a finite thick generalised hexagon or octagon with $G leqslant{rm Aut}(mathcal{S})$ acting point-primitively and the socle of $G$ isomorphic to ${rm PSL}_n(q)$ where $n geqslant 2$, then the stabiliser of a point acts irreducibly on the natural module. We describe a strategy to prove that such a generalised hexagon or octagon $mathcal{S}$ does not exist.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79212770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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