关于4p阶3度Cayley图的正规性的一点注记

群G的Cayley图Cay(G,S)称为是正规的,如果G的右正则表示R(G)在Cay(G,S)的全自同构群中正规.设p为奇素数,相关文献决定了4p阶连通3度Cayley图的正规性.本文给出了上述文献的主要结果的一个新的简短的证明.     

关于轮网络的一簇猜想

轮网络是由Cayley图模型设计出来的一种新型互连网络模型.星网络、冒泡排序网络、修正冒泡排序网络可嵌入轮网络.为了揭示它的整体结构,对轮网络提出如下一簇猜想:轮网络是边不交的i个Hamilton圈及2(n-i)-2个完美匹配的并,其中1≤i≤(n-1);并证明了当n=4,5,6,1≤i≤3时,猜想成立.     

有限群的子群完备码（英文）

本文从子群双陪集的角度给出了有限群的一般子群可作为子群完备码的充要条件.在此基础上给出了拟二面体群中的所有子群完备码.     

关于Sumner-Blitch猜想的一个注记

设G是一个图。G的最小度，连通度，控制数，独立控制数和独立数分别用δ，k，γ，i和α表示，图G是3－γ-临界的，如果γ=3，而且G增加任一条边所得的图的控制数为2.Sumner和Blitch猜想：任意连通的3－γ临界图满足i=3，本文证明了如果G是使α＝k 1≤δ的连通3－γ-临界图，那么Sumner-Blitch猜想成立。     

交错群A5的Cayley图同构的Li-Praeger猜想

设G是有限群, S是G＼{1}的子集,并满足S=S -1. 用X=Cay(G, S )表示G关于S的Cayley图. 称S为G的CI-子集, 如果对任意同构Cay(G, S )Cay(G, T )存在α∈Aut(G), 使得Sα=T .设m是正整数,称G为m-CI-群, 如果G的每个满足S =S -1和|S|≤m的子集S都是CI的. 证明了Li-Praeger猜想：交错群A₅是4- CI-群.     

关于超Euler图的一个猜想的注记

本文研究了Catlin的关于超Euler图的一个猜想,借助于收缩方法,得到了该猜想的两个充分条件.     

关于Freiman-Lev猜想的一些注记（英文）

本文关注Freiman-Lev的一个猜想,刻画了一些小限制和集的结构.     

关于M-R公开问题的注记

在本文中,通过关联ＢＣＫ-代数与立体格公理系统的联系,给出了Ｍ-Ｒ公开问题一个肯定的答复．     

A Conjecture Concerning a Limit of Non-Cayley Graphs

Our aim in this note is to present a transitive graph that we conjecture is not quasi-isometric to any Cayley graph. No such graph is currently known. Our graph arises both as an abstract limit in a suitable space of graphs and in a concrete way as a subset of a product of trees.     

On the Characterization of Linear Uniquely Decodable Codes

A Uniquely Decodable (UD) Code is a code such that any vector of the ambient space has a unique closest codeword. In this paper we begin a study of the structure of UD codes and identify perfect subcodes. In particular we determine all linear UD codes of covering radius 2.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ⁻¹. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ⁻¹ and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Zassenhaus猜想的一个注记

在这篇注记中,通过研究两个有限群直积的整群环的正规化挠单位的偏增广,证明了在某些条件下这些正规化挠单位与该群中的元素在有理群代数中共轭.     

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.     

Perfect state transfer in unitary Cayley graphs and gcd-graphs

In this work, we consider the problem on the existence of perfect state transfer in unitary Cayley graphs and gcd-graphs over finite commutative rings. We characterize all finite commutative rings allowing perfect transfer to occur on their unitary Cayley graphs. Also, we use our main result to study perfect state transfer in unitary Cayley signed graphs and some gcd-graphs on quotient rings of unique factorization domains. Moreover, we can apply some calculations on the eigenvalues to determine the existence of perfect state transfer in Cayley graphs over finite chain rings.     

Finding minimum dominating cycles in permutation graphs

A dominating cycle for a graph G = (V, E) is a subset C of V which has the following properties: (i) the subgraph of G induced by C has a Hamiltonian cycle, and (ii) every vertex of V is adjacent to some vertex of C. In this paper, we develop an O(n²) algorithm for finding a minimum cardinality dominating cycle in a permutation graph. We also show that a minimum cardinality dominating cycle in a permutation graph always has an even number of vertices unless it is isomorphic to C₃.