A full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order.
{"title":"Classification of minimal blocking sets in small Desarguesian projective planes","authors":"Kris Coolsaet, Arne Botteldoorn, Veerle Fack","doi":"10.1002/jcd.21842","DOIUrl":"https://doi.org/10.1002/jcd.21842","url":null,"abstract":"<p>A full classification (up to equivalence) of all minimal blocking sets in Desarguesian projective planes of order <math>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>≤</mo>\u0000 \u0000 <mn>8</mn>\u0000 </mrow>\u0000 </mrow></math> was obtained by computer. The resulting numbers of minimal blocking sets are tabulated according to size of the set and order of the automorphism group. For the minimal blocking sets with the larger automorphism groups explicit descriptions are given. Some of these results can also be generalised to Desarguesian projective planes of higher order.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 8","pages":"561-580"},"PeriodicalIF":0.7,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72170554","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}
G. Araujo-Pardo, R. Jajcay, Alejandra Ramos-Rivera, T. Szonyi
A bipartite biregular (m,n;g) $(m,n;g)$ ‐graph Γ ${rm{Gamma }}$ is a bipartite graph of even girth g $g$ having the degree set {m,n} ${m,n}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An (m,n;g) $(m,n;g)$ ‐bipartite biregular cage is a bipartite biregular (m,n;g) $(m,n;g)$ ‐graph of minimum order. In their 2019 paper, Filipovski, Ramos‐Rivera, and Jajcay present lower bounds on the orders of bipartite biregular (m,n;g) $(m,n;g)$ ‐graphs, and call the graphs that attain these bounds bipartite biregular Moore cages. In our paper, we improve the lower bounds obtained in the above paper. Furthermore, in parallel with the well‐known classical results relating the existence of k $k$ ‐regular Moore graphs of even girths g=6,8 $g=6,8$ , and 12 to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of an S(2,k,v) $S(2,k,v)$ ‐Steiner system yields the existence of a bipartite biregular k,v−1k−1;6 $left(k,frac{v-1}{k-1};6right)$ ‐cage, and, vice versa, the existence of a bipartite biregular (k,n;6) $(k,n;6)$ ‐cage whose order is equal to one of our lower bounds yields the existence of an S(2,k,1+n(k−1)) $S(2,k,1+n(k-1))$ ‐Steiner system. Moreover, with regard to the special case of Steiner triple systems, we completely solve the problem of determining the orders of (3,n;6) $(3,n;6)$ ‐bipartite biregular cages for all integers n≥4 $nge 4$ . Considering girths higher than 6, we relate the existence of generalized polygons (quadrangles, hexagons, and octagons) to the existence of (n+1,n2+1;8) $(n+1,{n}^{2}+1;8)$ ‐, (n2+1,n3+1;8) $({n}^{2}+1,{n}^{3}+1;8)$ ‐, (n,n+2;8) $(n,n+2;8)$ ‐, (n+1,n3+1;12) $(n+1,{n}^{3}+1;12)$ ‐ and (n+1,n2+1;16) $(n+1,{n}^{2}+1;16)$ ‐bipartite biregular cages, respectively. Using this connection, we also derive improved upper bounds for the orders of other classes of bipartite biregular cages of girths 8, 12, and 14.
{"title":"On a relation between bipartite biregular cages, block designs and generalized polygons","authors":"G. Araujo-Pardo, R. Jajcay, Alejandra Ramos-Rivera, T. Szonyi","doi":"10.1002/jcd.21836","DOIUrl":"https://doi.org/10.1002/jcd.21836","url":null,"abstract":"A bipartite biregular (m,n;g) $(m,n;g)$ ‐graph Γ ${rm{Gamma }}$ is a bipartite graph of even girth g $g$ having the degree set {m,n} ${m,n}$ and satisfying the additional property that the vertices in the same partite set have the same degree. An (m,n;g) $(m,n;g)$ ‐bipartite biregular cage is a bipartite biregular (m,n;g) $(m,n;g)$ ‐graph of minimum order. In their 2019 paper, Filipovski, Ramos‐Rivera, and Jajcay present lower bounds on the orders of bipartite biregular (m,n;g) $(m,n;g)$ ‐graphs, and call the graphs that attain these bounds bipartite biregular Moore cages. In our paper, we improve the lower bounds obtained in the above paper. Furthermore, in parallel with the well‐known classical results relating the existence of k $k$ ‐regular Moore graphs of even girths g=6,8 $g=6,8$ , and 12 to the existence of projective planes, generalized quadrangles, and generalized hexagons, we prove that the existence of an S(2,k,v) $S(2,k,v)$ ‐Steiner system yields the existence of a bipartite biregular k,v−1k−1;6 $left(k,frac{v-1}{k-1};6right)$ ‐cage, and, vice versa, the existence of a bipartite biregular (k,n;6) $(k,n;6)$ ‐cage whose order is equal to one of our lower bounds yields the existence of an S(2,k,1+n(k−1)) $S(2,k,1+n(k-1))$ ‐Steiner system. Moreover, with regard to the special case of Steiner triple systems, we completely solve the problem of determining the orders of (3,n;6) $(3,n;6)$ ‐bipartite biregular cages for all integers n≥4 $nge 4$ . Considering girths higher than 6, we relate the existence of generalized polygons (quadrangles, hexagons, and octagons) to the existence of (n+1,n2+1;8) $(n+1,{n}^{2}+1;8)$ ‐, (n2+1,n3+1;8) $({n}^{2}+1,{n}^{3}+1;8)$ ‐, (n,n+2;8) $(n,n+2;8)$ ‐, (n+1,n3+1;12) $(n+1,{n}^{3}+1;12)$ ‐ and (n+1,n2+1;16) $(n+1,{n}^{2}+1;16)$ ‐bipartite biregular cages, respectively. Using this connection, we also derive improved upper bounds for the orders of other classes of bipartite biregular cages of girths 8, 12, and 14.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"6 1","pages":"479 - 496"},"PeriodicalIF":0.7,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81848714","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}
We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper subsystem which contains it. We give sharp upper bounds on the size of a minimal spreading set in a Steiner triple system and show that if all the minimal spreading sets are large then the examined triple system must be a projective space. We also show that the size of a minimal spreading set is not an invariant of a Steiner triple system.
{"title":"Steiner triple systems and spreading sets in projective spaces","authors":"Zoltán Lóránt Nagy, Levente Szemerédi","doi":"10.1002/jcd.21841","DOIUrl":"https://doi.org/10.1002/jcd.21841","url":null,"abstract":"<p>We address several extremal problems concerning the spreading property of point sets of Steiner triple systems. This property is closely related to the structure of subsystems, as a set is spreading if and only if there is no proper subsystem which contains it. We give sharp upper bounds on the size of a minimal spreading set in a Steiner triple system and show that if all the minimal spreading sets are large then the examined triple system must be a projective space. We also show that the size of a minimal spreading set is not an invariant of a Steiner triple system.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 8","pages":"549-560"},"PeriodicalIF":0.7,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21841","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72162623","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}
We propose a method of constructing block designs which combine genetic algorithm and a method for constructing designs with a prescribed automorphism group using tactical decompositions (i.e., orbit matrices). We apply this method to construct new Steiner systems with parameters and new symmetric designs with parameters (71, 15, 3).
{"title":"Constructing block designs with a prescribed automorphism group using genetic algorithm","authors":"Dean Crnković, Tin Zrinski","doi":"10.1002/jcd.21838","DOIUrl":"https://doi.org/10.1002/jcd.21838","url":null,"abstract":"<p>We propose a method of constructing block designs which combine genetic algorithm and a method for constructing designs with a prescribed automorphism group using tactical decompositions (i.e., orbit matrices). We apply this method to construct new Steiner systems with parameters <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>5</mn>\u0000 <mo>,</mo>\u0000 <mn>45</mn>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $S(2,5,45)$</annotation>\u0000 </semantics></math> and new symmetric designs with parameters (71, 15, 3).</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 7","pages":"515-526"},"PeriodicalIF":0.7,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72161518","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}
We propose a method of constructing block designs which combine genetic algorithm and a method for constructing designs with a prescribed automorphism group using tactical decompositions (i.e., orbit matrices). We apply this method to construct new Steiner systems with parameters S(2,5,45) $S(2,5,45)$ and new symmetric designs with parameters (71, 15, 3).
{"title":"Constructing block designs with a prescribed automorphism group using genetic algorithm","authors":"D. Crnković, Tin Zrinski","doi":"10.1002/jcd.21838","DOIUrl":"https://doi.org/10.1002/jcd.21838","url":null,"abstract":"We propose a method of constructing block designs which combine genetic algorithm and a method for constructing designs with a prescribed automorphism group using tactical decompositions (i.e., orbit matrices). We apply this method to construct new Steiner systems with parameters S(2,5,45) $S(2,5,45)$ and new symmetric designs with parameters (71, 15, 3).","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"5 1","pages":"515 - 526"},"PeriodicalIF":0.7,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80926054","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}
R. Julian R. Abel, Thomas Britz, Yudhistira A. Bunjamin, Diana Combe
In this paper we provide a 4-GDD of type , thereby solving the existence question for the last remaining feasible type for a 4-GDD with no more than 30 points. We then show that 4-GDDs of type exist for all but a finite specified set of feasible pairs .
在本文中，我们提供了一个类型为2 2 5 5${2}^{2}{5}^{5}$的4-GDD，从而解决了不超过30分的4-GDD的最后一个剩余可行类型的存在性问题。然后我们证明了类型为2 t 5 s$｛2｝^｛t｝｛5｝^｝s｝$的4-GDD存在于除有限的指定可行对集之外的所有可行对（t，s）$（t，s）$。
{"title":"Group divisible designs with block size 4 and group sizes 2 and 5","authors":"R. Julian R. Abel, Thomas Britz, Yudhistira A. Bunjamin, Diana Combe","doi":"10.1002/jcd.21830","DOIUrl":"https://doi.org/10.1002/jcd.21830","url":null,"abstract":"<p>In this paper we provide a 4-GDD of type <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 \u0000 <mn>2</mn>\u0000 </msup>\u0000 \u0000 <msup>\u0000 <mn>5</mn>\u0000 \u0000 <mn>5</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${2}^{2}{5}^{5}$</annotation>\u0000 </semantics></math>, thereby solving the existence question for the last remaining feasible type for a 4-GDD with no more than 30 points. We then show that 4-GDDs of type <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 \u0000 <mi>t</mi>\u0000 </msup>\u0000 \u0000 <msup>\u0000 <mn>5</mn>\u0000 \u0000 <mi>s</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation> ${2}^{t}{5}^{s}$</annotation>\u0000 </semantics></math> exist for all but a finite specified set of feasible pairs <math>\u0000 \u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>t</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>s</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(t,s)$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 6","pages":"367-383"},"PeriodicalIF":0.7,"publicationDate":"2022-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21830","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72173929","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}
Let K v ( 3 ) = ( X , ℰ ) ${K}_{v}^{(3)}=(X,{rm{ {mathcal E} }})$ be the complete hypergraph, uniform of rank 3, defined on a vertex set X = { x 1 , … , x v } $X={{x}_{1},ldots ,{x}_{v}}$ , so that ℰ ${rm{ {mathcal E} }}$ is the set of all triples of X $X$ . Let H ( 3 ) = ( V , D ) ${H}^{(3)}=(V,{mathscr{D}})$ be a subhypergraph of K v ( 3 ) ${K}_{v}^{(3)}$ , which means that V ⊆ X $Vsubseteq X$ and D ⊆ ℰ ${mathscr{D}}subseteq {rm{ {mathcal E} }}$ . We call 3‐edges the triples of V $V$ contained in the family D ${mathscr{D}}$ and edges the pairs of V $V$ contained in the 3‐edges of D ${mathscr{D}}$ , that we denote by [ x , y ] $[x,y]$ . A H ( 3 ) ${H}^{(3)}$ ‐design Σ ${rm{Sigma }}$ is called edge balanced if for any x , y ∈ X $x,yin X$ , x ≠ y $xne y$ , the number of blocks of Σ ${rm{Sigma }}$ containing the edge [ x , y ] $[x,y]$ is constant. In this paper, we consider the star hypergraph S ( 3 ) ( 2 , m + 2 ) ${S}^{(3)}(2,m+2)$ , which is a hypergraph with m $m$ 3‐edges such that all of them have an edge in common. We completely determine the spectrum of edge balanced S ( 3 ) ( 2 , m + 2 ) ${S}^{(3)}(2,m+2)$ ‐designs for any m ≥ 2 $mge 2$ , that is, the set of the orders v $v$ for which such a design exists. Then we consider the case m = 2 $m=2$ and we denote the hypergraph S ( 3 ) ( 2 , 4 ) ${S}^{(3)}(2,4)$ by P ( 3 ) ( 2 , 4 ) ${P}^{(3)}(2,4)$ . Starting from any edge‐balanced S ( 3 ) 2 , v + 4 3 ${S}^{(3)}left(2,frac{v+4}{3}right)$ , with v ≡ 2 mod 3 $vequiv 2,mathrm{mod},3$ sufficiently big, for any p ∈ N $pin {mathbb{N}}$ , v 2 ≤ p ≤ v $unicode{x02308}frac{v}{2}unicode{x02309}le ple v$ , we construct a P ( 3 ) ( 2 , 4 ) ${P}^{(3)}(2,4)$ ‐design of order 2 v $2v$ with feasible set { 2 , 3 } ∪ [ p , v ] ${2,3}cup [p,v]$ , in the context of proper vertex colorings such that no block is either monochromatic or polychromatic.
{"title":"Edge balanced star‐hypergraph designs and vertex colorings of path designs","authors":"Paola Bonacini, Lucia Marino","doi":"10.1002/jcd.21837","DOIUrl":"https://doi.org/10.1002/jcd.21837","url":null,"abstract":"Let K v ( 3 ) = ( X , ℰ ) ${K}_{v}^{(3)}=(X,{rm{ {mathcal E} }})$ be the complete hypergraph, uniform of rank 3, defined on a vertex set X = { x 1 , … , x v } $X={{x}_{1},ldots ,{x}_{v}}$ , so that ℰ ${rm{ {mathcal E} }}$ is the set of all triples of X $X$ . Let H ( 3 ) = ( V , D ) ${H}^{(3)}=(V,{mathscr{D}})$ be a subhypergraph of K v ( 3 ) ${K}_{v}^{(3)}$ , which means that V ⊆ X $Vsubseteq X$ and D ⊆ ℰ ${mathscr{D}}subseteq {rm{ {mathcal E} }}$ . We call 3‐edges the triples of V $V$ contained in the family D ${mathscr{D}}$ and edges the pairs of V $V$ contained in the 3‐edges of D ${mathscr{D}}$ , that we denote by [ x , y ] $[x,y]$ . A H ( 3 ) ${H}^{(3)}$ ‐design Σ ${rm{Sigma }}$ is called edge balanced if for any x , y ∈ X $x,yin X$ , x ≠ y $xne y$ , the number of blocks of Σ ${rm{Sigma }}$ containing the edge [ x , y ] $[x,y]$ is constant. In this paper, we consider the star hypergraph S ( 3 ) ( 2 , m + 2 ) ${S}^{(3)}(2,m+2)$ , which is a hypergraph with m $m$ 3‐edges such that all of them have an edge in common. We completely determine the spectrum of edge balanced S ( 3 ) ( 2 , m + 2 ) ${S}^{(3)}(2,m+2)$ ‐designs for any m ≥ 2 $mge 2$ , that is, the set of the orders v $v$ for which such a design exists. Then we consider the case m = 2 $m=2$ and we denote the hypergraph S ( 3 ) ( 2 , 4 ) ${S}^{(3)}(2,4)$ by P ( 3 ) ( 2 , 4 ) ${P}^{(3)}(2,4)$ . Starting from any edge‐balanced S ( 3 ) 2 , v + 4 3 ${S}^{(3)}left(2,frac{v+4}{3}right)$ , with v ≡ 2 mod 3 $vequiv 2,mathrm{mod},3$ sufficiently big, for any p ∈ N $pin {mathbb{N}}$ , v 2 ≤ p ≤ v $unicode{x02308}frac{v}{2}unicode{x02309}le ple v$ , we construct a P ( 3 ) ( 2 , 4 ) ${P}^{(3)}(2,4)$ ‐design of order 2 v $2v$ with feasible set { 2 , 3 } ∪ [ p , v ] ${2,3}cup [p,v]$ , in the context of proper vertex colorings such that no block is either monochromatic or polychromatic.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 1","pages":"497 - 514"},"PeriodicalIF":0.7,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80950113","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}
<p>We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the concurrence matrix of a PGD can have at most three distinct eigenvalues, all of which are nonnegative integers. The matrix contains useful information on the incidence structure of the design. An ordinary 2-<math> <semantics> <mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> $(v,k,lambda )$</annotation> </semantics></math> design has a single concurrence <math> <semantics> <mrow> <mrow> <mi>λ</mi> </mrow> </mrow> <annotation> $lambda $</annotation> </semantics></math>, and its concurrence matrix is circulant, a partial geometry has two concurrences 1 and 0, and a transversal design <math> <semantics> <mrow> <mrow> <msub> <mtext>TD</mtext> <mi>λ</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>,</mo> <mi>u</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${text{TD}}_{lambda }(k,u)$</annotation> </semantics></math> has two concurrences <math> <semantics> <mrow> <mrow> <mi>λ</mi> </mrow> </mrow> <annotation> $lambda $</annotation> </semantics></math> and 0, and its concurrence matrix is circulant. In this paper, we survey the known PGDs by highlighting their concurrences and constructio
{"title":"Partial geometric designs having circulant concurrence matrices","authors":"Sung-Yell Song, Theodore Tranel","doi":"10.1002/jcd.21834","DOIUrl":"https://doi.org/10.1002/jcd.21834","url":null,"abstract":"<p>We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the concurrence matrix of a PGD can have at most three distinct eigenvalues, all of which are nonnegative integers. The matrix contains useful information on the incidence structure of the design. An ordinary 2-<math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>λ</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(v,k,lambda )$</annotation>\u0000 </semantics></math> design has a single concurrence <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math>, and its concurrence matrix is circulant, a partial geometry has two concurrences 1 and 0, and a transversal design <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mtext>TD</mtext>\u0000 \u0000 <mi>λ</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>u</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${text{TD}}_{lambda }(k,u)$</annotation>\u0000 </semantics></math> has two concurrences <math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $lambda $</annotation>\u0000 </semantics></math> and 0, and its concurrence matrix is circulant. In this paper, we survey the known PGDs by highlighting their concurrences and constructio","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 6","pages":"420-460"},"PeriodicalIF":0.7,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jcd.21834","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72142712","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}
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