Journal of Combinatorial Designs最新文献_第6页

Generalizations of some Nordhaus–Gaddum-type results on spectral radius 谱半径上某些Nordhaus–Gaddum型结果的推广

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2023-09-21 DOI: 10.1002/jcd.21919

Junying Lu, Lanchao Wang, Yaojun Chen

{"title":"Generalizations of some Nordhaus–Gaddum-type results on spectral radius","authors":"Junying Lu, Lanchao Wang, Yaojun Chen","doi":"10.1002/jcd.21919","DOIUrl":"https://doi.org/10.1002/jcd.21919","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> be a simple graph and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $lambda (G)$</annotation>\u0000 </semantics></math> the spectral radius of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. For <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $kge 2$</annotation>\u0000 </semantics></math>, a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-edge decomposition <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>H</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>…</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mi>H</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $({H}_{1},{rm{ldots }},{H}_{k})$</annotation>\u0000 </semantics></math> is <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> spanning subgraphs such that their edge sets form a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math>-partition of the edge set of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math>. In this paper, we obtain some sharp lower and upper bounds for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 ","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 12","pages":"701-712"},"PeriodicalIF":0.7,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50153173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Construction of rank 4 self-dual association schemes inducing three partial geometric designs 诱导三个局部几何设计的秩为4的自对偶关联方案的构造

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2023-09-05 DOI: 10.1002/jcd.21917

Akihide Hanaki

引用次数: 0

proper partial geometries with an automorphism group acting primitively on points and lines 自同构群初始作用于点和线的适当局部几何

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2023-08-31 DOI: 10.1002/jcd.21914

Wendi Di

{"title":"proper partial geometries with an automorphism group acting primitively on points and lines","authors":"Wendi Di","doi":"10.1002/jcd.21914","DOIUrl":"https://doi.org/10.1002/jcd.21914","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{S}}$</annotation>\u0000 </semantics></math> be a finite proper partial geometry pg<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>t</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>α</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(s,t,alpha )$</annotation>\u0000 </semantics></math> not isomorphic to the van Lint–Schrijver partial geometry pg<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>5</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>5</mn>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(5,5,2)$</annotation>\u0000 </semantics></math> and let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> be a group of automorphisms of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{S}}$</annotation>\u0000 </semantics></math> acting primitively on both points and lines of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation> ${mathscr{S}}$</annotation>\u0000 </semantics></math>, we show that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>60</mn>\u0000 </mrow>\u0000 <annotation> $alpha le 60$</annotation>\u0000 </semantics></math> then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> must be almost simple.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 11","pages":"642-664"},"PeriodicalIF":0.7,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50148812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","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 directed Oberwolfach problem for complete symmetric equipartite digraphs and uniform-length cycles 关于完全对称等分有向图和一致长环的有向Oberwolfach问题

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2023-08-22 DOI: 10.1002/jcd.21913

Nevena Francetić, Mateja Šajna

{"title":"On the directed Oberwolfach problem for complete symmetric equipartite digraphs and uniform-length cycles","authors":"Nevena Francetić, Mateja Šajna","doi":"10.1002/jcd.21913","DOIUrl":"https://doi.org/10.1002/jcd.21913","url":null,"abstract":"We examine the necessary and sufficient conditions for a complete symmetric equipartite digraph <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>K</mi>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mrow>\u0000 <mo>[</mo>\u0000 \u0000 <mi>m</mi>\u0000 \u0000 <mo>]</mo>\u0000 </mrow>\u0000 </mrow>\u0000 \u0000 <mo>*</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation> ${K}_{n[m]}^{* }$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math> parts of size <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 </mrow>\u0000 <annotation> $m$</annotation>\u0000 </semantics></math> to admit a resolvable decomposition into directed cycles of length <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>. We show that the obvious necessary conditions are sufficient for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>t</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation> $m,n,tge 2$</annotation>\u0000 </semantics></math> in each of the following four cases: (i) <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $m(n-1)$</annotation>\u0000 </semantics></math> is even; (ii) <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>gcd</mtext>\u0000 \u0000 <mrow>\u0000 <mo>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 11","pages":"604-641"},"PeriodicalIF":0.7,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21913","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50140497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A generalization of group divisible t $t$ -designs 群可整除t$t$-设计的一个推广

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2023-08-10 DOI: 10.1002/jcd.21912

Sijia Liu, Yue Han, Lijun Ma, Lidong Wang, Zihong Tian

{"title":"A generalization of group divisible \u0000 \u0000 \u0000 t\u0000 \u0000 $t$\u0000 -designs","authors":"Sijia Liu, Yue Han, Lijun Ma, Lidong Wang, Zihong Tian","doi":"10.1002/jcd.21912","DOIUrl":"https://doi.org/10.1002/jcd.21912","url":null,"abstract":"Cameron defined the concept of generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs, which generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs, resolvable designs and orthogonal arrays. This paper introduces a new class of combinatorial designs which simultaneously provide a generalization of both generalized <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs and group divisible <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs. In certain cases, we derive necessary conditions for the existence of generalized group divisible <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs, and then point out close connections with various well-known classes of designs, including mixed orthogonal arrays, factorizations of the complete multipartite graphs, large sets of group divisible designs, and group divisible designs with (orthogonal) resolvability. Moreover, we investigate constructions for generalized group divisible <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 </mrow>\u0000 <annotation> $t$</annotation>\u0000 </semantics></math>-designs and almost completely determine their existence for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation> $t=2,3$</annotation>\u0000 </semantics></math> and small block sizes.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 11","pages":"575-603"},"PeriodicalIF":0.7,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50137077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Totally symmetric quasigroups of order 16 16阶全对称拟群

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2023-07-13 DOI: 10.1002/jcd.21910

Hy Ginsberg

引用次数: 0

Enumerating Steiner triple systems Steiner三重系统的枚举

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2023-07-13 DOI: 10.1002/jcd.21906

Daniel Heinlein, Patric R. J. Östergård

引用次数: 0

The chromatic index of finite projective spaces 有限射影空间的色指数

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2023-06-30 DOI: 10.1002/jcd.21904

Lei Xu, Tao Feng

{"title":"The chromatic index of finite projective spaces","authors":"Lei Xu, Tao Feng","doi":"10.1002/jcd.21904","DOIUrl":"https://doi.org/10.1002/jcd.21904","url":null,"abstract":"A line coloring of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(n,q)$</annotation>\u0000 </semantics></math>, the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-dimensional projective space over GF<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>q</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation> $(q)$</annotation>\u0000 </semantics></math>, is an assignment of colors to all lines of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(n,q)$</annotation>\u0000 </semantics></math> so that any two lines with the same color do not intersect. The chromatic index of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>q</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $text{PG}(n,q)$</annotation>\u0000 </semantics></math>, denoted by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 \u0000 <mo>′</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mtext>PG</mtext>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"432-446"},"PeriodicalIF":0.7,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50148157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Linear and circular single-change covering designs revisited 重新审视线性和圆形单次变更覆盖设计

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2023-06-01 DOI: 10.1002/jcd.21885

Amanda Chafee, Brett Stevens

{"title":"Linear and circular single-change covering designs revisited","authors":"Amanda Chafee, Brett Stevens","doi":"10.1002/jcd.21885","DOIUrl":"https://doi.org/10.1002/jcd.21885","url":null,"abstract":"A single-change covering design (SCCD) is a <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math>-set <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math> and an ordered list <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℒ</mi>\u0000 </mrow>\u0000 <annotation> ${rm{ {mathcal L} }}$</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $b$</annotation>\u0000 </semantics></math> blocks of size <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $k$</annotation>\u0000 </semantics></math> where every pair from <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <annotation> $X$</annotation>\u0000 </semantics></math> must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>,</mo>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation> $v,k$</annotation>\u0000 </semantics></math>. In this paper, we use a new recursive construction to solve the existence of circular SCCD(<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>,</mo>\u0000 <mn>4</mn>\u0000 <mo>,</mo>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $v,4,b$</annotation>\u0000 </semantics></math>) for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 </mrow>\u0000 <annotation> $v$</annotation>\u0000 </semantics></math> and three residue classes of circular SCCD(<math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>v</mi>\u0000 <mo>,</mo>\u0000 <mn>5</mn>\u0000 <mo>,</mo>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation> $v,5,b$</annotation>\u0000 </semantics","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"405-421"},"PeriodicalIF":0.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21885","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50116119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Self-dual association schemes, fusions of Hamming schemes, and partial geometric designs 自对偶关联方案、Hamming方案的融合和部分几何设计

IF 0.7 4区数学

Journal of Combinatorial Designs Pub Date : 2023-05-25 DOI: 10.1002/jcd.21889

Bangteng Xu

{"title":"Self-dual association schemes, fusions of Hamming schemes, and partial geometric designs","authors":"Bangteng Xu","doi":"10.1002/jcd.21889","DOIUrl":"https://doi.org/10.1002/jcd.21889","url":null,"abstract":"Partial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will first study the properties and characterizations of self-dual association schemes. Then using the characterizations of self-dual association schemes and the representation theory (character tables) of commutative association schemes, we obtain characterizations and classifications of self-dual (symmetric or nonsymmetric) association schemes of rank 4 that produce as many as possible nontrivial partial geometric designs or 2-designs. In particular, for a primitive self-dual symmetric association scheme <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>X</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <msub>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <msub>\u0000 <mi>R</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mo>}</mo>\u0000 </mrow>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>i</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${mathscr{X}}=(X,{{{R}_{i}}}_{0le ile 3})$</annotation>\u0000 </semantics></math> of rank 4, if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∣</mo>\u0000 \u0000 <mi>X</mi>\u0000 \u0000 <mo>∣</mo>\u0000 </mrow>\u0000 <annotation> $| X| $</annotation>\u0000 </semantics></math> is a power of 3 and each of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>R</mi>\u0000 \u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation> $","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 8","pages":"373-399"},"PeriodicalIF":0.7,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50120463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1