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.     

Perfect state transfer on bi-Cayley graphs over abelian groups

The study of perfect state transfer on graphs has attracted a great deal of attention during the past ten years because of its applications to quantum information processing and quantum computation. Perfect state transfer is understood to be a rare phenomenon. This paper establishes necessary and sufficient conditions for a bi-Cayley graph having a perfect state transfer over any given finite abelian group. As corollaries, many known and new results are obtained on Cayley graphs having perfect state transfer over abelian groups, (generalized) dihedral groups, semi-dihedral groups and generalized quaternion groups. Especially, we give an example of a connected non-normal Cayley graph over a dihedral group having perfect state transfer between two distinct vertices, which was thought impossible.     

关于n-可扩图的一些结果

本文讨论了关于m-可扩图的两个极值问题；并考查了下述图类的n-可扩性；正则偶图，单位区间图和分裂图。     

Tetravalent Edge-transitive Cayley Graphs of PGL （2, p）

t Let F = Cay（G, S）, R（G） be the right regular representation of G. The graph Г is called normal with respect to G, if R（G） is normal in the full automorphism group Aut（F） of F. Г is called a bi-normal with respect to G if R（G） is not normal in Aut（Г）, but R（G） contains a subgroup of index 2 which is normal in Aut（F）. In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL（2,p） are either normal or bi-normal when p ≠ 11 is a prime.     

On the extendability of Bi-Cayley graphs of finite abelian groups

Let G be a finite group and A a nonempty subset (possibly containing the identity element) of G. The Bi-Cayley graph X=BC(G,A) of G with respect to A is defined as the bipartite graph with vertex set G×{0,1} and edge set {{(g,0),(sg,1)}∣g∈G,s∈A}. A graph Γ admitting a perfect matching is called n-extendable if ∣V(Γ)∣≥2n+2 and every matching of size n in Γ can be extended to a perfect matching of Γ. In this paper, the extendability of Bi-Cayley graphs of finite abelian groups is explored. In particular, 2-extendable and 3-extendable Bi-Cayley graphs of finite abelian groups are characterized.     

Subgroup regular sets in Cayley graphs

Let Γ be a graph with vertex set V, and let a and b be nonnegative integers. A subset C of V is called an $(a,b)$-regular set in Γ if every vertex in C has exactly a neighbors in C and every vertex in $V?C$ has exactly b neighbors in C. In particular, $(0,1)$-regular sets and $(1,1)$-regular sets in Γ are called perfect codes and total perfect codes in Γ, respectively. A subset C of a group G is said to be an $(a,b)$-regular set of G if there exists a Cayley graph of G which admits C as an $(a,b)$-regular set. In this paper we prove that, for any generalized dihedral group G or any group G of order 4p or pq for some primes p and q, if a nontrivial subgroup H of G is a $(0,1)$-regular set of G, then it must also be an $(a,b)$-regular set of G for any $0?a?|H|?1$ and $0?b?|H|$ such that a is even when $|H|$ is odd. A similar result involving $(1,1)$-regular sets of such groups is also obtained in the paper.     

4度Cayley图的Hamilton圈分解的新方法与理论证明

给出了＂Hamilton圈侧枝循环＂等四个定理.它揭示了Abel群上4度Cayley图的Hamilton圈分解的特点及规律.同时,提出了Hamilton圈上＂单向通道＂的＂离合＂理论.在此基础上给出了Abel群上4度Cayley图的Hamilton圈分解的新方法-＂离合法＂,此方法具有简明、快捷、分解方案多的特点.另外,Hamilton圈＂单向通道＂的＂离合＂理论还为解决6度Cayley图的Hamilton圈分解奠定了理论基础.     

p~2阶弧传递循环图的正规性条件

群G关于S的有向Cayley图X=Cay(G,S)称为pk阶有向循环图,若G是pk阶循环群.利用有限群论和图论的较深刻的结果,对p2阶弧传递(有向)循环图的正规性条件进行了讨论,证明了任一p2阶弧传递(有向)循环图是正规的当且仅当(|Aut(G,S)|,p)=1.     

Perfect State Transfer on Weighted Abelian Cayley Graphs*

Recently, there are extensive studies on perfect state transfer (PST for short) on graphs due to their significant applications in quantum information processing and quantum computations. However, there is not any general characterization of graphs that have PST in literature. In this paper, the authors present a depiction on weighted abelian Cayley graphs having PST. They give a unified approach to describe the periodicity and the existence of PST on some specific graphs.     

Balanced Cayley graphs and balanced planar graphs

A balanced graph is a bipartite graph with no induced circuit of length . These graphs arise in integer linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for balanced regular graphs are also presented. The graphs in this paper are simple.