{"title":"On a certain class of 1-thin distance-regular graphs","authors":"Mark S. MacLean, Štefko Miklavič","doi":"10.26493/1855-3974.2193.0b0","DOIUrl":"https://doi.org/10.26493/1855-3974.2193.0b0","url":null,"abstract":"Let Γ denote a non-bipartite distance-regular graph with vertex set X , diameter D ≥ 3 , and valency k ≥ 3 . Fix x ∈ X and let T = T ( x ) denote the Terwilliger algebra of Γ with respect to x . For any z ∈ X and for 0 ≤ i ≤ D , let Γ i ( z ) = { w ∈ X : ∂( z , w ) = i }. For y ∈ Γ 1 ( x ) , abbreviate D j i = D j i ( x , y ) = Γ i ( x ) ∩ Γ j ( y ) (0 ≤ i , j ≤ D ) . For 1 ≤ i ≤ D and for a given y , we define maps H i : D i i → ℤ and V i : D i − 1 i ∪ D i i − 1 → ℤ as follows: H i ( z ) = | Γ 1 ( z ) ∩ D i − 1 i − 1 |, V i ( z ) = | Γ 1 ( z ) ∩ D i − 1 i − 1 |. We assume that for every y ∈ Γ 1 ( x ) and for 2 ≤ i ≤ D , the corresponding maps H i and V i are constant, and that these constants do not depend on the choice of y . We further assume that the constant value of H i is nonzero for 2 ≤ i ≤ D . We show that every irreducible T -module of endpoint 1 is thin. Furthermore, we show Γ has exactly three irreducible T -modules of endpoint 1, up to isomorphism, if and only if three certain combinatorial conditions hold. As examples, we show that the Johnson graphs J ( n , m ) where n ≥ 7, 3 ≤ m < n /2 satisfy all of these conditions.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90241225","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":"Hypergeometric degenerate Bernoulli polynomials and numbers","authors":"T. Komatsu","doi":"10.26493/1855-3974.1907.3C2","DOIUrl":"https://doi.org/10.26493/1855-3974.1907.3C2","url":null,"abstract":"Carlitz defined the degenerate Bernoulli polynomials β n ( λ , x ) by means of the generating function t ((1 + λ t ) 1/ λ − 1) −1 (1 + λ t ) x / λ . In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In this paper, we show some expressions and properties of hypergeometric degenerate Bernoulli polynomials β N , n ( λ , x ) and numbers, in particular, in terms of determinants. The coefficients of the polynomial β n ( λ , 0) were completely determined by Howard in 1996. We determine the coefficients of the polynomial β N , n ( λ , 0) . Hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers appear in the coefficients.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73730987","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":"Well-totally-dominated graphs","authors":"Selim Bahadır, T. Ekim, Didem Gözüpek","doi":"10.26493/1855-3974.2465.571","DOIUrl":"https://doi.org/10.26493/1855-3974.2465.571","url":null,"abstract":"A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating set. In this paper, we study graphs whose all minimal total dominating sets have the same size, referred to as well-totally-dominated (WTD) graphs. We first show that WTD graphs with bounded total domination number can be recognized in polynomial time. Then we focus on WTD graphs with total domination number two. In this case, we characterize triangle-free WTD graphs and WTD graphs with packing number two, and we show that there are only finitely many planar WTD graphs with minimum degree at least three. Lastly, we show that if the minimum degree is at least three then the girth of a WTD graph is at most 12. We conclude with several open questions.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73759102","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 the incidence maps of incidence structures","authors":"Tim Penttila, A. Siciliano","doi":"10.26493/1855-3974.1996.DB7","DOIUrl":"https://doi.org/10.26493/1855-3974.1996.DB7","url":null,"abstract":"By using elementary linear algebra methods we exploit properties of the incidence map of certain incidence structures with finite block sizes. We give new and simple proofs of theorems of Kantor and Lehrer, and their infinitary version. Similar results are obtained also for diagrams geometries. By mean of an extension of Block’s Lemma on the number of orbits of an automorphism group of an incidence structure, we give informations on the number of orbits of: a permutation group (of possible infinite degree) on subsets of finite size; a collineation group of a projective and affine space (of possible infinite dimension) over a finite field on subspaces of finite dimension; a group of isometries of a classical polar space (of possible infinite rank) over a finite field on totally isotropic subspaces (or singular in case of orthogonal spaces) of finite dimension. Furthermore, when the structure is finite and the associated incidence matrix has full rank, we give an alternative proof of a result of Camina and Siemons. We then deduce that certain families of incidence structures have no sharply transitive sets of automorphisms acting on blocks.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83637655","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":"The distinguishing index of connected graphs without pendant edges","authors":"W. Imrich, R. Kalinowski, M. Pilsniak, M. Wozniak","doi":"10.26493/1855-3974.1852.4f7","DOIUrl":"https://doi.org/10.26493/1855-3974.1852.4f7","url":null,"abstract":"We consider edge colourings, not necessarily proper. The distinguishing index D ′( G ) of a graph G is the least number of colours in an edge colouring that is preserved only by the identity automorphism. It is known that D ′( G ) ≤ Δ for every countable, connected graph G with finite maximum degree Δ except for three small cycles. We prove that D ′( G ) ≤ ⌈√Δ⌉ + 1 if additionally G does not have pendant edges.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80827857","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":"Notes on exceptional signed graphs","authors":"Z. Stanić","doi":"10.26493/1855-3974.1933.2DF","DOIUrl":"https://doi.org/10.26493/1855-3974.1933.2DF","url":null,"abstract":"A connected signed graph is called exceptional if it has a representation in the root system E 8 , but has not in any D k . In this study we obtain some properties of these signed graphs, mostly expressed in terms of those that are maximal with a fixed number of eigenvalues distinct from −2 . As an application, we characterize exceptional signed graphs with exactly 2 eigenvalues. In some particular cases, we prove the (non-)existence of such signed graphs.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75841855","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}
Gang Chen, Jiawei He, Ilia N. Ponomarenko, A. Vasil’ev
{"title":"A characterization of exceptional pseudocyclic association schemes by multidimensional intersection numbers","authors":"Gang Chen, Jiawei He, Ilia N. Ponomarenko, A. Vasil’ev","doi":"10.26493/1855-3974.2405.B43","DOIUrl":"https://doi.org/10.26493/1855-3974.2405.B43","url":null,"abstract":"Recent classification of $frac{3}{2}$-transitive permutation groups leaves us with three infinite families of groups which are neither $2$-transitive, nor Frobenius, nor one-dimensional affine. The groups of the first two families correspond to special actions of ${mathrm{PSL}}(2,q)$ and ${mathrm{PGamma L}}(2,q),$ whereas those of the third family are the affine solvable subgroups of ${mathrm{AGL}}(2,q)$ found by D. Passman in 1967. The association schemes of the groups in each of these families are known to be pseudocyclic. It is proved that apart from three particular cases, each of these exceptional pseudocyclic schemes is characterized up to isomorphism by the tensor of its $3$-dimensional intersection numbers.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76996606","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":"Generalized dihedral CI-groups","authors":"Ted Dobson, M. Muzychuk, Pablo Spiga","doi":"10.26493/1855-3974.2443.02e","DOIUrl":"https://doi.org/10.26493/1855-3974.2443.02e","url":null,"abstract":"In this paper, we find a strong new restriction on the structure of CI-groups. We show that, if $R$ is a generalised dihedral group and if $R$ is a CI-group, then for every odd prime $p$ the Sylow $p$-subgroup of $R$ has order $p$, or $9$. Consequently, any CI-group with quotient a generalised dihedral group has the same restriction, that for every odd prime $p$ the Sylow $p$-subgroup of the group has order $p$, or $9$. We also give a counter example to the conjecture that every BCI-group is a CI-group.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87062870","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 2-closures of rank 3 groups","authors":"S. Skresanov","doi":"10.26493/1855-3974.2450.1DC","DOIUrl":"https://doi.org/10.26493/1855-3974.2450.1DC","url":null,"abstract":"A permutation group $G$ on $Omega$ is called a rank 3 group if it has precisely three orbits in its induced action on $Omega times Omega$. The largest permutation group on $Omega$ having the same orbits as $G$ on $Omega times Omega$ is called the 2-closure of $G$. A description of 2-closures of rank 3 groups is given. As a special case, it is proved that 2-closure of a primitive one-dimensional affine rank 3 permutation group of sufficiently large degree is also affine and one-dimensional.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77116550","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":"New results on modular Golomb rulers, optical orthogonal codes and related structures","authors":"M. Buratti, Douglas R Stinson","doi":"10.26493/1855-3974.2374.9ff","DOIUrl":"https://doi.org/10.26493/1855-3974.2374.9ff","url":null,"abstract":"We prove new existence and nonexistence results for modular Golomb rulers in this paper. We completely determine which modular Golomb rulers of order $k$ exist, for all $kleq 11$, and we present a general existence result that holds for all $k geq 3$. We also derive new nonexistence results for infinite classes of modular Golomb rulers and related structures such as difference packings, optical orthogonal codes, cyclic Steiner systems and relative difference families.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87586996","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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