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On Roli's cube 在罗利的立方体上
Art Discret. Appl. Math. Pub Date : 2021-02-17 DOI: 10.26493/2590-9770.1411.6EE
B. Monson
{"title":"On Roli's cube","authors":"B. Monson","doi":"10.26493/2590-9770.1411.6EE","DOIUrl":"https://doi.org/10.26493/2590-9770.1411.6EE","url":null,"abstract":"First described in 2014, Roli’s cube ℛ is a chiral 4-polytope, faithfully realized in Euclidean 4-space (a situation earlier thought to be impossible). Here we describe ℛ in a new way, determine its minimal regular cover, and reveal connections to the Mobius-Kantor configuration.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123943197","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 12-regular nut graphs 在12正则坚果图上
Art Discret. Appl. Math. Pub Date : 2021-02-08 DOI: 10.26493/2590-9770.1403.1B1
N. Bašić, M. Knor, R. Škrekovski
{"title":"On 12-regular nut graphs","authors":"N. Bašić, M. Knor, R. Škrekovski","doi":"10.26493/2590-9770.1403.1B1","DOIUrl":"https://doi.org/10.26493/2590-9770.1403.1B1","url":null,"abstract":"A nut graph is a simple graph whose adjacency matrix is singular with 1-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each d ∈ {3, 4, …, 11} all values n such that there exists a d-regular nut graph of order n. In the present paper, we resolve the first open case d = 12, i.e. we show that there exists a 12-regular nut graph of order n if and only if n ≥ 16. We also present a result by which there are infinitely many circulant nut graphs of degree d ≡ 0 (mod  4) and no circulant nut graphs of degree d ≡ 2 (mod  4). The former result partially resolves a question by Fowler et al. on existence of vertex-transitive nut graphs of order n and degree d. We conclude the paper with problems, conjectures and ideas for further work.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126359931","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: 6
Minimizing vertex-degree function index for k-generalized quasi-unicyclic graphs k-广义拟单环图顶点度函数索引的最小化
Art Discret. Appl. Math. Pub Date : 2021-02-05 DOI: 10.26493/2590-9770.1364.48B
I. Tomescu
{"title":"Minimizing vertex-degree function index for k-generalized quasi-unicyclic graphs","authors":"I. Tomescu","doi":"10.26493/2590-9770.1364.48B","DOIUrl":"https://doi.org/10.26493/2590-9770.1364.48B","url":null,"abstract":"In this paper the problem of minimizing vertex-degree function index Hf(G) for k-generalized quasi-unicyclic graphs of order n is solved for k ≥ 1 and n ≥ 2k + 2 if the function f is strictly increasing and strictly convex. These conditions are fulfilled by general first Zagreb index 0Rα(G) if α > 1, second multiplicative Zagreb index ∏2(G) and sum lordeg index SL(G). The extremal graph is unique for k = 1, n = 4 and for k ≥ 2 and it consists from a path x1, x2, …, xn − 1 and a new vertex xn adjacent with xk, xk + 1 and xk + 2.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122655365","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 the direct products of skew-morphisms 关于斜态射的直积
Art Discret. Appl. Math. Pub Date : 2021-02-01 DOI: 10.26493/2590-9770.1388.C56
Junyang Zhang
{"title":"On the direct products of skew-morphisms","authors":"Junyang Zhang","doi":"10.26493/2590-9770.1388.C56","DOIUrl":"https://doi.org/10.26493/2590-9770.1388.C56","url":null,"abstract":"A skew-morphism φ of a finite group G is a permutation on G fixing the identity element of G and for which there is an integer-valued function π on G such that φ(gh) = φ(g)φπ(g)(h) for all g, h ∈ G. For two permutations α : A → A and β : B → B on the sets A and B, their direct product α × β is the permutation on the Cartesian product A × B given by (α × β)(a, b) = (α(a), β(b)) for all (a, b) ∈ A × B. In this paper, necessary and sufficient conditions for a direct product of two skew-morphisms to still be a skew-morphism are given.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"114 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116393279","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
Whitney's connectivity inequalities for directed hypergraphs 有向超图的Whitney连通性不等式
Art Discret. Appl. Math. Pub Date : 2021-01-30 DOI: 10.26493/2590-9770.1380.1C9
Anahy Santiago Arguello, P. Stadler
{"title":"Whitney's connectivity inequalities for directed hypergraphs","authors":"Anahy Santiago Arguello, P. Stadler","doi":"10.26493/2590-9770.1380.1C9","DOIUrl":"https://doi.org/10.26493/2590-9770.1380.1C9","url":null,"abstract":"Whitney’s inequality established an important connection between vertex and edge connectivity and the degree of a graph, which was later generalized to digraphs and to undirected hypergraphs. Here we show, using the most common definitions of connectedness for directed hypergraphs, that an analogous result holds directed hypergraphs. It relates the vertex connectivity under strong vertex elimination, edge connectivity under weak edge elimination, and a suitable degree-like parameter and is a proper generalization of the situation in both digraphs and undirected hypergraphs. We furthermore relate the connectivity parameters of directed hypergraphs with those of its directed bipartite Konig representation.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"24 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133169441","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
A simple construction of exponentially many nonisomorphic orientable triangular embeddings of K_12s K_12s的指数多非同构可定向三角形嵌入的简单构造
Art Discret. Appl. Math. Pub Date : 2021-01-26 DOI: 10.26493/2590-9770.1387.A84
V. P. Korzhik
{"title":"A simple construction of exponentially many nonisomorphic orientable triangular embeddings of K_12s","authors":"V. P. Korzhik","doi":"10.26493/2590-9770.1387.A84","DOIUrl":"https://doi.org/10.26493/2590-9770.1387.A84","url":null,"abstract":"Using an index one current graph with the cyclic current group we give a simple construction of 22s − 7 nonisomorphic orientable triangular embeddings of the complete graph K12s, s ≥ 4. These embeddings have no nontrivial automorphisms.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131869293","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
Semi-perimeter and inner site-perimeter of k-ary words and bargraphs k元字和图的半周长和内点周长
Art Discret. Appl. Math. Pub Date : 2021-01-24 DOI: 10.26493/2590-9770.1376.A29
T. Mansour
{"title":"Semi-perimeter and inner site-perimeter of k-ary words and bargraphs","authors":"T. Mansour","doi":"10.26493/2590-9770.1376.A29","DOIUrl":"https://doi.org/10.26493/2590-9770.1376.A29","url":null,"abstract":"Given a bargraph B, a border cell of B is a cell of B that shares at least one common edge with an outside cell of B. Clearly, the inner site-perimeter of B is the number of border cells of B. A tangent cell of B is a cell of B which is not a border cell of B and shares at least one vertex with an outside cell of B. In this paper, we study the generating function for the number of k-ary words, represented as bargraphs, according to the number of horizontal steps, up steps, border cells and tangent cells. This allows us to express some cases via Chebyshev polynomials of the second kind. Moreover, we find an explicit formula for the number of bargraphs according to the number of horizontal steps, up steps, and tangent cells/inner site-perimeter. We also derive asymptotic estimates for the mean number of tangent cells/inner site-perimeter.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127846684","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
A note on fractional covers of a graph 对图的分数覆盖的注释
Art Discret. Appl. Math. Pub Date : 2020-12-20 DOI: 10.26493/2590-9770.1398.fe9
John Baptist Gauci, J. P. Zerafa
{"title":"A note on fractional covers of a graph","authors":"John Baptist Gauci, J. P. Zerafa","doi":"10.26493/2590-9770.1398.fe9","DOIUrl":"https://doi.org/10.26493/2590-9770.1398.fe9","url":null,"abstract":"A fractional colouring of a graph $G$ is a function that assigns a non-negative real value to all possible colour-classes of $G$ containing any vertex of $G$, such that the sum of these values is at least one for each vertex. The fractional chromatic number is the minimum sum of the values assigned by a fractional colouring over all possible such colourings of $G$. Introduced by Bosica and Tardif, fractional covers are an extension of fractional colourings whereby the real-valued function acts on all possible subgraphs of $G$ belonging to a given class of graphs. The fractional chromatic number turns out to be a special instance of the fractional cover number. In this work we investigate fractional covers acting on $(k+1)$-clique-free subgraphs of $G$ which, although sharing some similarities with fractional covers acting on $k$-colourable subgraphs of $G$, they exhibit some peculiarities. We first show that if a simple graph $G_2$ is a homomorphic image of a simple graph $G_1$, then the fractional cover number defined on the $(k+1)$-clique-free subgraphs of $G_1$ is bounded above by the corresponding number of $G_2$. We make use of this result to obtain bounds for the associated fractional cover number of graphs that are either $n$-colourable or $a!!:!!b$-colourable.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114195211","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
Classifying edge-biregular maps of negative prime Euler characteristic 分类负素数欧拉特征的边双正则映射
Art Discret. Appl. Math. Pub Date : 2020-10-31 DOI: 10.26493/2590-9770.1392.f9a
Olivia Reade, J. Širáň
{"title":"Classifying edge-biregular maps of negative prime Euler characteristic","authors":"Olivia Reade, J. Širáň","doi":"10.26493/2590-9770.1392.f9a","DOIUrl":"https://doi.org/10.26493/2590-9770.1392.f9a","url":null,"abstract":"An edge-biregular map arises as a smooth normal quotient of a unique index-two subgroup of a full triangle group acting with two edge-orbits. We give a classification of all finite edge-biregular maps on surfaces of negative prime Euler characteristic.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123628107","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
Topology of clique complexes of line graphs 线形图团复形的拓扑
Art Discret. Appl. Math. Pub Date : 2020-09-25 DOI: 10.26493/2590-9770.1434.bf4
Shuchita Goyal, Samir Shukla, Anurag Singh
{"title":"Topology of clique complexes of line graphs","authors":"Shuchita Goyal, Samir Shukla, Anurag Singh","doi":"10.26493/2590-9770.1434.bf4","DOIUrl":"https://doi.org/10.26493/2590-9770.1434.bf4","url":null,"abstract":"Clique complex of a line graph is a functor from the category of graphs to the category of simplicial complexes. Using functorial properties of this functor, we determine the homotopy type of clique complexes of line graphs for several classes of graphs. Among others, we study triangle free graphs, wheel free graphs, 4-regular circulant graphs, chordal graphs, and complete multipartite graphs. We also give a closed form formula for the homotopy type of these complexes in several cases.","PeriodicalId":236892,"journal":{"name":"Art Discret. Appl. Math.","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131323949","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: 4
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