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An alternate proof of the monotonicity of the number of positive entries in nonnegative matrix powers 非负矩阵幂中正项数单调性的一个备用证明
Art Discret. Appl. Math. Pub Date : 2020-08-11 DOI: 10.26493/2590-9770.1280.4DA
S. Filipovski
{"title":"An alternate proof of the monotonicity of the number of positive entries in nonnegative matrix powers","authors":"S. Filipovski","doi":"10.26493/2590-9770.1280.4DA","DOIUrl":"https://doi.org/10.26493/2590-9770.1280.4DA","url":null,"abstract":"Let A be a nonnegative real matrix of order n and f(A) denote the number of positive entries in A. In 2018, Xie proved that if f(A) ≤ 3 or f(A) ≥ n2 − 2n + 2, then the sequence (f(Ak))k = 1∞ is monotone for positive integers k. In this note we give an alternate proof of this result by counting walks in a digraph of order n.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128306276","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}
引用次数: 0
On 2-skeleta of hypercubes
Art Discret. Appl. Math. Pub Date : 2020-08-10 DOI: 10.26493/2590-9770.1302.f4e
P. C. Kainen
{"title":"On 2-skeleta of hypercubes","authors":"P. C. Kainen","doi":"10.26493/2590-9770.1302.f4e","DOIUrl":"https://doi.org/10.26493/2590-9770.1302.f4e","url":null,"abstract":"It is shown that the 2-skeleton of the odd-d-dimensional hypercube can be decomposed into sd spheres and τd tori, where sd = (d − 1)2d − 4 and τd is asymptotically in the range (64/9)2d − 7 to (d − 1)(d − 3)2d − 7.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133607062","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}
引用次数: 3
On the Terwilliger algebra of certain family of bipartite distance-regular graphs with Δ_2 = 0 关于Δ_2 = 0的二部距离正则图族的Terwilliger代数
Art Discret. Appl. Math. Pub Date : 2020-08-10 DOI: 10.26493/2590-9770.1271.e54
Štefko Miklavič, Safet Penjić
{"title":"On the Terwilliger algebra of certain family of bipartite distance-regular graphs with Δ_2 = 0","authors":"Štefko Miklavič, Safet Penjić","doi":"10.26493/2590-9770.1271.e54","DOIUrl":"https://doi.org/10.26493/2590-9770.1271.e54","url":null,"abstract":"Let Γ denote a bipartite distance-regular graph with diameter D ≥ 4 and valency k ≥ 3. Let X denote the vertex set of Γ, and let Ai (0 ≤ i ≤ D) denote the distance matrices of Γ. We abbreviate A := A1. For x ∈ X and for 0 ≤ i ≤ D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. \u0000Fix x ∈ X and let T = T(x) denote the subalgebra of MatX(ℂ) generated by A, E0*, E1*, …, ED*, where for 0 ≤ i ≤ D, Ei* represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T-module W we mean min{i ∣ Ei*W ≠ 0}. \u0000In this paper we assume Γ has the property that for 2 ≤ i ≤ D − 1, there exist complex scalars αi, βi such that for all y, z ∈ X with ∂(x, y) = 2, ∂(x, z) = i, ∂(y, z) = i, we have αi + βi|Γ1(x) ∩ Γ1(y) ∩ Γi − 1(z)| = |Γi − 1(x) ∩ Γi − 1(y) ∩ Γ1(z)|. \u0000We study the structure of irreducible T-modules of endpoint 2. Let W denote an irreducible T-module with endpoint 2, and let v denote a nonzero vector in E2*W. We show that W = span({Ei*Ai − 2E2*v ∣ 2 ≤ i ≤ D} ∪ {Ei*Ai + 2E2*v ∣ 2 ≤ i ≤ D − 2}). \u0000It turns out that, except for a particular family of bipartite distance-regular graphs with D = 5, this result is already known in the literature. Assume now that Γ is a member of this particular family of graphs. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2 and it is not thin. We give a basis for this T-module.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134491992","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}
引用次数: 2
Hereditary polyhedra with planar regular faces 具有平面正面的遗传多面体
Art Discret. Appl. Math. Pub Date : 2020-08-10 DOI: 10.26493/2590-9770.1281.0ad
A. I. Weiss, E. Schulte
{"title":"Hereditary polyhedra with planar regular faces","authors":"A. I. Weiss, E. Schulte","doi":"10.26493/2590-9770.1281.0ad","DOIUrl":"https://doi.org/10.26493/2590-9770.1281.0ad","url":null,"abstract":"A skeletal polyhedron in Euclidean 3-space is called hereditary if the symmetries of each face extend to symmetries of the entire polyhedron. In this paper we describe the finite hereditary skeletal polyhedra which have regular convex polygons or regular starpolygons as faces.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123841452","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}
引用次数: 1
Reflexible complete regular dessins and antibalanced skew morphisms of cyclic groups 环群的完全正则解和反平衡斜态
Art Discret. Appl. Math. Pub Date : 2020-08-03 DOI: 10.26493/2590-9770.1284.3ad
Kan Hu, Young Soo Kwon
{"title":"Reflexible complete regular dessins and antibalanced skew morphisms of cyclic groups","authors":"Kan Hu, Young Soo Kwon","doi":"10.26493/2590-9770.1284.3ad","DOIUrl":"https://doi.org/10.26493/2590-9770.1284.3ad","url":null,"abstract":"A skew morphism of a finite group A is a bijection φ on A fixing the identity element of A and for which there exists an integer-valued function π on A such that φ(ab) = φ(a)φ(b), for all a, b ∈ A. In addition, if φ−1(a) = φ(a−1)−1, for all a ∈ A, then φ is called antibalanced. In this paper we develop a general theory of antibalanced skew morphisms and establish a one-to-one correspondence between reciprocal pairs of antibalanced skew morphisms of the cyclic additive groups and isomorphism classes of reflexible regular dessins with complete bipartite underlying graphs. As an application, reflexible complete regular dessins are classified.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"290 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114940433","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}
引用次数: 1
A new generalization of generalized Petersen graphs 广义Petersen图的新推广
Art Discret. Appl. Math. Pub Date : 2020-07-27 DOI: 10.26493/2590-9770.1279.02c
Katarína Jasenčáková, R. Jajcay, T. Pisanski
{"title":"A new generalization of generalized Petersen graphs","authors":"Katarína Jasenčáková, R. Jajcay, T. Pisanski","doi":"10.26493/2590-9770.1279.02c","DOIUrl":"https://doi.org/10.26493/2590-9770.1279.02c","url":null,"abstract":"We discuss a new family of cubic graphs, which we call group divisible generalised Petersen graphs (GDGP -graphs), that bears a close resemblance to the family of generalised Petersen graphs, both in definition and properties. The focus of our paper is on determining the algebraic properties of graphs from our new family. We look for highly symmetric graphs, e.g., graphs with large automorphism groups, and vertexor arc-transitive graphs. In particular, we present arithmetic conditions for the defining parameters that guarantee that graphs with these parameters are vertex-transitive or Cayley, and we find one arctransitive GDGP -graph which is neither a CQ graph of Feng and Wang, nor a generalised Petersen graph.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130844009","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}
引用次数: 3
Regular balanced Cayley maps on nonabelian metacyclic groups of odd order 奇阶非abel元环群上的正则平衡Cayley映射
Art Discret. Appl. Math. Pub Date : 2020-07-27 DOI: 10.26493/2590-9770.1290.f73
Kai Yuan, Yan Wang, H. Qu
{"title":"Regular balanced Cayley maps on nonabelian metacyclic groups of odd order","authors":"Kai Yuan, Yan Wang, H. Qu","doi":"10.26493/2590-9770.1290.f73","DOIUrl":"https://doi.org/10.26493/2590-9770.1290.f73","url":null,"abstract":"In this paper, we show that nonabelian metacyclic groups of odd order do not have regular balanced Cayley maps.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"168 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123277617","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}
引用次数: 1
On strongly sequenceable abelian groups 强序列阿贝尔群
Art Discret. Appl. Math. Pub Date : 2020-07-22 DOI: 10.26493/2590-9770.1291.c54
B. Alspach, Georgina Liversidge
{"title":"On strongly sequenceable abelian groups","authors":"B. Alspach, Georgina Liversidge","doi":"10.26493/2590-9770.1291.c54","DOIUrl":"https://doi.org/10.26493/2590-9770.1291.c54","url":null,"abstract":"A group is strongly sequenceable if every connected Cayley digraph on the group admits an orthogonal directed cycle or an orthogonal directed path. This paper deals with the problem of whether finite abelian groups are strongly sequenceable. A method based on posets is used to show that if the connection set for a Cayley digraph on an abelian group has cardinality at most nine, then the digraph admits either an orthogonal directed path or an orthogonal directed cycle.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123964986","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}
引用次数: 20
Two new families of non-CCA groups 两个非cca团体的新家族
Art Discret. Appl. Math. Pub Date : 2020-05-24 DOI: 10.26493/2590-9770.1370.445
Brandon Fuller, Joy Morris
{"title":"Two new families of non-CCA groups","authors":"Brandon Fuller, Joy Morris","doi":"10.26493/2590-9770.1370.445","DOIUrl":"https://doi.org/10.26493/2590-9770.1370.445","url":null,"abstract":"We determine two new infinite families of Cayley graphs that admit colour-preserving automorphisms that do not come from the group action. By definition, this means that these Cayley graphs fail to have the CCA (Cayley Colour Automorphism) property, and the corresponding infinite families of groups also fail to have the CCA property. The families of groups consist of the direct product of any dihedral group of order $2n$ where $n ge 3$ is odd, with either itself, or the cyclic group of order $n$. In particular, this family of examples includes the smallest non-CCA group that had not fit into any previous family of known non-CCA groups.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122620574","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}
引用次数: 0
Groups for which it is easy to detect graphical regular representations 易于检测图形规则表示的组
Art Discret. Appl. Math. Pub Date : 2020-05-20 DOI: 10.26493/2590-9770.1373.60A
D. Morris, Joy Morris, Gabriel Verret
{"title":"Groups for which it is easy to detect graphical regular representations","authors":"D. Morris, Joy Morris, Gabriel Verret","doi":"10.26493/2590-9770.1373.60A","DOIUrl":"https://doi.org/10.26493/2590-9770.1373.60A","url":null,"abstract":"We say that a finite group G is \"DRR-detecting\" if, for every subset S of G, either the Cayley digraph Cay(G,S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism phi of G such that phi(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product of two cyclic groups of order p is not DRR-detecting, for every odd prime p. We also show that if G and H are nontrivial groups that admit a digraphical regular representation and either gcd(|G|,|H|) = 1, or H is not DRR-detecting, then the direct product G x H is not DRR-detecting. Some of these results also have analogues for graphical regular representations.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"121 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128613952","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}
引用次数: 1
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