距离正则图的推广（英文）

本文研究了直径为d(Γ)≥2的距离正则图Γ的补图.利用Γ的交叉数分别证明了当d=2时,Γ的补图式强正则;当d≥3时,Γ的补图是广义强正则.将文献[2]中的距离正则图Grassmann图、对偶极图、Hamming图推广到它们的补图,从而得到广义强正则图.     

具有广义可行性余弦序列的E1?Ed型距离正则图

给出了具有广义可行性余弦序列的E₁?E_d型距离正则图的特征,并计算了这类图的交叉数.     

4K＋1阶强正则自补图的存在唯一性

４ｋ＋１阶强正则自补图的存在唯一性刘象武（哈尔滨师范大学数学系哈尔滨１５００８０）关键词：循环图；强正则自补图；原根ＡＭＳ（１９９１）主题分类：０５Ｅ３０．本文所述的图均指有限无向简单图．本文分别用字母户，庄，人和产表示强正则图的顶点数、正则度、两相...     

二倍无平方因子阶的3度1-正则图

一个图叫做1-正则的, 如果它的自同构群在它的弧集上作用正则. 设n是一个无平方因子的正整数. 证明了存在2n阶3度1-正则图当且仅当n=3^tp₁p₂… p_s≥13, 其中t≤1, s≥1, p_i (1≤ i≤s)为互不相同的素数且满足3|(p_i-1). 进一步, 对每个满足上述条件的整数n, 共有2^s−1个互不同构的2n阶3度1-正则图, 并且这些图均为2n阶二面体群上的Cayley图. 由此可知, 不存在4m阶3度1-正则图, 其中m为无平方因子的奇数.     

强P-正则半群的一个构造

本文研究了强P-正则半群的结构.利用正则*-半群和一族满足某种条件的映射给出了强P-正则半群的一个构造,推广和丰富了相关文献中纯正半群的结果.     

二部图的距离k次方和问题

本文定义S_k（G）为G中所有点对之间距离的k次方之和.利用顶点划分的方法得到了直径为d的n顶点连通二部图S_k（G）的下界,并确定了达到下界所对应的的极图.     

一类2级一般强正则图

作为强正则图的一种新推广,p级一般强正则图是使得任意两个邻接的顶点和任意两个不邻接的顶点的公共邻接点数都有p种不同取值的非空k-正则图.对于参数为(n,k;a1,a2;c1,c2)的2级一般强正则图中任一顶点,如果与这个顶点邻接且有ai(i=1,2)个公共邻接点或者与这个顶点不邻接且有ci(i=1,2)个公共邻接点的顶...     

Zn[i] 的平方映射图

本文研究了模n 高斯整数环Z_n[i] 的平方映射图Γ(n). 利用数论、图论与群论等方法, 获得了Γ(n) 中顶点0 及1 的入度, 并研究了Γ(n) 的零因子子图的半正则性. 同时, 获得了Γ(n) 中顶点的高度公式.推广了Somer 等人给出的模n 剩余类环平方映射图的相关结论.     

极小正则2-图及其应用

2-图是边的尺寸至多为2的超图,极小正则2-图是不含有真正则因子的正则2-图. 设f₂(n)为所有n个顶点的极小正则2-图的最大度数.给出了极小正则2-图的一个结构性质，并由此证得 f₂(n) =(n+3-i)/3, 其中1≤i≤6, n≥7, i≡n(mod 6),从而解决了范红兵等人提出的一个猜想. 作为在图论中的应用, 可以刻画不可分解因子的正则图, 并给出关于度条件的最好可能的因子存在性定理. 进而, f₂(n)和极小2-图可应用于最初引发这项研究的通用开关盒设计问题.     

有限赋值AR-箭图导出的Lie代数

设Γ是连通赋值AR-箭图,用￡(Γ)=_x∈Γ0Zu_x表示由Γ的顶点集Γ₀生成的自由Abel群,^~Γ为Γ的泛覆盖,基本群为G,证明了当Γ是有限连通的赋值AR-箭图时,￡(Γ)关于括号运算作成(Γ)₁的Lie子代数且￡(Γ)/G (Γ)。这里 (Γ)₁是Γ的退化Hall代数,(^~Γ)/G是由^~Γ导出的轨道Lie代数。     

Strongly Regular Decompositions of the Complete Graph

We study several questions about amorphic association schemes and other strongly regular decompositions of the complete graph. We investigate how two commuting edge-disjoint strongly regular graphs interact. We show that any decomposition of the complete graph into three strongly regular graphs must be an amorphic association scheme. Likewise we show that any decomposition of the complete graph into strongly regular graphs of (negative) Latin square type is an amorphic association scheme. We study strongly regular decompositions of the complete graph consisting of four graphs, and find a primitive counterexample to A.V. Ivanov's conjecture which states that any association scheme consisting of strongly regular graphs only must be amorphic.     

Distance-Regular Graphs with Strongly Regular Subconstituents

In [3] Cameron et al. classified strongly regular graphs with strongly regular subconstituents. Here we prove a theorem which implies that distance-regular graphs with strongly regular subconstituents are precisely the Taylor graphs and graphs with a ₁ = 0 and a _i {0,1} for i = 2,...,d.     

On Strongly Regular Graphs with k=2μ and Their Extensions

We obtain a convenient expression for the parameters of a strongly regular graph with k=2 in terms of the nonprincipal eigenvalues x and –y. It turns out in particular that such graphs are pseudogeometric for pG _x(2x,y–1). We prove that a strongly regular graph with parameters (35,16,6,8) is a quotient of the Johnson graph (8,4). We also find the parameters of strongly regular graphs in which the neighborhoods of vertices are pseudogeometric graphs for pG _x(2x,t),x3. In consequence, we establish that a connected graph in which the neighborhoods of vertices are pseudogeometric graphs for pG ₃(6,2) is isomorphic to the Taylor graph on 72 vertices or to the alternating form graph Alt(4,2) with parameters (64,35,18,20).     

Strongly regular Cayley graphs and rationality of relative Gauss sums

In this note, we give a construction of strongly regular Cayley graphs. The presented construction is based on choosing cyclotomic classes in finite fields, and our results generalize ten of the eleven sporadic examples of cyclotomic strongly regular graphs given by Schmidt and White [B. Schmidt, C. White, All two-weight irreducible cyclic codes, Finite Fields Appl. 8 (2002), 321–367] into infinite families. These infinite families of strongly regular graphs have new parameters. The main tools that we employed are relative Gauss sums instead of explicit evaluations of Gauss sums.     

Lifting constructions of strongly regular Cayley graphs

We give two “lifting” constructions of strongly regular Cayley graphs. In the first construction we “lift” a cyclotomic strongly regular graph by using a subdifference set of the Singer difference sets. The second construction uses quadratic forms over finite fields and it is a common generalization of the construction of the affine polar graphs [7] and a construction of strongly regular Cayley graphs given in [15]. The two constructions are related in the following way: the second construction can be viewed as a recursive construction, and the strongly regular Cayley graphs obtained from the first construction can serve as starters for the second construction. We also obtain association schemes from the second construction.     

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     

Restrictions on parameters of partial difference sets in nonabelian groups

A partial difference set $S$ in a finite group $G$ satisfying $1?S$ and $S=S^{?1}$ corresponds to an undirected strongly regular Cayley graph $\text{Cay}\left(G,S\right)$. While the case when $G$ is abelian has been thoroughly studied, there are comparatively few results when $G$ is nonabelian. In this paper, we provide restrictions on the parameters of a partial difference set that apply to both abelian and nonabelian groups and are especially effective in groups with a nontrivial center. In particular, these results apply to $p$‐groups, and we are able to rule out the existence of partial difference sets in many instances.     

A note on triangle‐free quasi‐symmetric designs

Triangle‐free quasi‐symmetric 2‐ (v, k, λ) designs with intersection numbers x, y; 0<x<y<kand λ>1, are investigated. It is proved that λ?2y ? x ? 3. As a consequence it is seen that for fixed λ, there are finitely many triangle‐free quasi‐symmetric designs. It is also proved that: k?y(y ? x) + x. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:422‐426, 2011     

Cyclotomy and Strongly Regular Graphs

We consider strongly regular graphs defined on a finite field by taking the union of some cyclotomic classes as difference set. Several new examples are found.