The Regular Near Polygons of Order (s, 2)

In this note we classify the regular near polygons of order (s, 2).     

Bounding the size of near hexagons with lines of size 3

We consider finite near hexagons with lines of size 3, and prove that there are only finitely many examples given the nondegeneracy condition that for each point there exists a point at distance 3 to it. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 108–122, 2006     

The completion of the classification of the regular near octagons with thick quads

Brouwer and Wilbrink [3] showed the nonexistence of regular near octagons whose parameters s, t₂, t₃ and t satisfy s ≥ 2, t₂ ≥ 2 and t₃ ≠ t₂(t₂+1). Later an arithmetical error was discovered in the proof. Because of this error, the existence problem was still open for the near octagons corresponding with certain values of s, t₂ and t₃. In the present paper, we will also show the nonexistence of these remaining regular near octagons. MSC2000 05B25, 05E30, 51E12 Postdoctoral Fellow of the Research Foundation - Flanders     

Valuations of glued near hexagons

Valuations of near polygons were introduced in [ 12 ] as an important tool for classifying dense near polygons. In the present article, we will introduce the class of the semi‐diagonal valuations. These valuations live in glued near hexagons. A glued near hexagon S can be coordinatized by a pair of admissible triples; such triples consist of a Steiner system , a group G, and a certain nice map . We will give a necessary and sufficient condition for the existence of semi‐diagonal valuations in in terms of these two admissible triples. For two classes of glued near hexagons, we will use this condition to determine all semi‐diagonal valuations. Each semi‐diagonal valuation will also give rise to a hyperplane of the glued near hexagon. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 35–48, 2007     

The Distance-2-Sets of the Slim Dense Near Hexagons

We determine all distance-2-sets and all valuations of the 11 slim dense near hexagons. We consider this as a necessary step in order to classify all slim dense near octagons. These near octagons will be classified in a forthcoming paper. Received November 29, 2004     

序为(3,3)的正则拟多边形

利用距离正则图的特征值方法,得到如下结论:设Γ是一个有序对为(3,3)正则拟多边形,如果d=r+1,则cd≠1,2,3.     

On Perfect 2‐Colorings of Johnson Graphs J(v, 3)

We study the perfect 2‐colorings (also known as the equitable partitions into two parts or the completely regular codes with covering radius 1) of the Johnson graphs . In particular, we classify all the realizable quotient matrices of perfect 2‐colorings for odd v. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 232–252, 2013     

一类(a+1,3)型的距离正则图

设Γ是直径为d且型为(α 1,3)的距离正则图,其中α≥2.用l(c,a,b)表示交叉阵列l(Γ)中列(c,a,b)~t的个数,记r=r(Γ)=l(c_1,a_1,b_1),s=s(Γ)=l(c_(r 1),a(r 1),b_(r 1)).那末,若c_(r 1)=3且a_(r 1)=3a,则d=r s 1,c_d=4且Γ为正则拟2d边形.     

A class of amply regular graphs related to the subconstituents of a dual polar graph

The connected components of the induced graphs on each subconstituent of the dual polar graph of the odd dimensional orthogonal spaces over a finite field are shown to be amply regular. The connected components of the graphs on the second and third subconstituents are shown to be distance-regular by elementary methods.     

SL(m,C)-equivariant and translation covariant continuous tensor valuations

The space of continuous, $\mathrm{SL}(m,\mathbb{C})$-equivariant, $m\ge 2$, and translation covariant valuations taking values in the space of real symmetric tensors on ${\mathbb{C}}^m?{\mathbb{R}}^{2m}$ of rank $r\ge 0$ is completely described. The classification involves the moment tensor valuation for $r\ge 1$ and is analogous to the known classification of the corresponding tensor valuations that are $\mathrm{SL}(2m,\mathbb{R})$-equivariant, although the method of proof cannot be adapted.     

链状正则图的平均距离

本文构造了一类链状正则图G_k∶δ,求出了它们的平均距离D(G_k.δ),并得到关系式上式等号成立当且仅当δ=4f且k=0.这个估计式指出了施容华猜想[1]D(G)≤n/(δ 1)不成立. 文中进一步证明了这一类链状正则图有最大的直径,所以可以作出猜想: 若G是n阶连通图,则D(G)<(n 1)/(δ 1),其中δ是图G的最小度。     

极大饱和图D（n,k）的极值性质及圈唯一性

本文证明了极大饱和图Ｄ（ｎ，ｋ）的一个极值性质：在与Ｄ（ｎ，ｋ）具有相同度序 所有图中，唯有Ｄ（ｎ，ｋ）含有最少的Ｋ３子图，并由此推出，在几乎正则图的范围内，ｋ个完全图之并及完全ｋ部多分图均是圈唯一的。本文还用图谱方法，证明了完全二分图Ｋｍ，ｎ的圈唯一性。     

The Non-Collinearity Graph of the O+(8,2) Quadric Is Uniquely Geometrisable

The graph of the titlehas the points of the O⁺(8,2) polar space as itsvertices, two such vertices being adjacent iff the correspondingpoints are non-collinear in the polar space. We prove that, uptoisomorphism, there is a unique partial geometry pg(8,7,4)whose point graph is this graph. This is the partial geometryof Cohen, Haemers and Van Lint and De Clerck, Dye and Thas. Ouruniqueness proof shows that this geometry has a subgeometry isomorphicto the affine plane of order three, and the geometry is canonicallydescribeable in terms of this affine plane.     

Off‐diagonal and asymptotic results on the Ramsey number r(K2,m,K2,n)

An upper bound on the Ramsey number r(K_2,n‐s,K_2,n) where s ≥ 2 is presented. Considering certain r(K_2,n‐s,K_2,n)‐colorings obtained from strongly regular graphs, we additionally prove that this bound matches the exact value of r(K_2,n‐s,K_2,n) in infinitely many cases if holds. Moreover, the asymptotic behavior of r(K_2,m,K_2,n) is studied for n being sufficiently large depending on m. We conclude with a table of all known Ramsey numbers r(K_2,m,K_2,n) where m,n ≤ 10. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 252–268, 2003     

n≤12阶(k,l)-正则极大平面图

我们知道当图的顶点数n>12时不存在正则极大平面图.相关文献提出了(k,l)-正则极大平面图的概念,并讨论了(5,6)-正则极大平面图的存在性.在相关文献中,作者分别讨论了阶n>12的(k,l)-正则极大平面图的存在条件及构造方法.本文讨论了阶n(≤12)的(k,l)-正则极大平面图的存在性,除两种情况外,本文给出了阶n(≤12)的(k,l)-正则极大平面图的存在条件及其一种构造的例子.     

Configuration distribution and designs of codes in the Johnson scheme

The main goal of this article is to present several connections between perfect codes in the Johnson scheme and designs, and provide new tools for proving Delsarte conjecture that there are no nontrivial perfect Codes in the Johnson scheme. Three topics will be considered. The first is the configuration distribution which is akin to the weight distribution in the Hamming scheme. We prove that if there exists an e‐perfect code in the Johnson scheme then there is a formula which connects the number of vectors at distance i from any codeword in various codes isomorphic to . The second topic is the Steiner systems embedded in a perfect code. We prove a lower bound on the number of Steiner systems embedded in a perfect code. The last topic is the strength of a perfect code. We show two new methods for computing the strength of a perfect code and demonstrate them on 1‐perfect codes. We further discuss how to settle Delsarte conjecture. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 15–34, 2007     

The Collinearity Graph of the O-(8,2) Quadric Is Not Geometrisable

The result of the title was conjectured by F. De Clerck and V. D. Tonchev (Partial geometries and Quadrics, Sankhya, special volume 54 (1992), 137–145). We prove it here.     

关于图的L(d,1)-标号问题

图G的L(2,1)-标号是一个从顶点集V(G)到非负整数集的函数f(x),使得若d(x,y)=1,则|f(x)-f(y)|≥2;若d(x,y)=2,则|f(x)-f(y)|≥1.图G的L(2,1)-标号数λ(G)是使得G有max{f(v)v∈V(G)}=k的L(2,1)-标号中的最小数k.Griggs和Yeh猜想对最大度为△的一般图G,有λ(G)≤△2.此文研究了作为L(2,1)-标号问题的推广的L(d,1)-标号问题,并得出了平面三角剖分图、立体四面体剖分图、平面近四边形剖分图的L(d,1)-标号的上界,作为推论证明了对上述几类图该猜想成立.