Arnbjörg Soffía Árnadóttir, Chris Godsil
In this paper, we characterize perfect state transfer in Cayley graphs for abelian groups that have a cyclic Sylow-2-subgroup. This generalizes a result of Bašić from 2013 where he provides a similar characterization for Cayley graphs of cyclic groups.
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Fractional revival, known as a quantum transport phenomenon, is essential for entanglement generation in quantum spin networks. The concept of fractional revival is a generalization of perfect state transfer and periodicity on graphs. In this paper, we propose a sufficient and necessary condition for abelian Cayley graphs having fractional revival between any two distinct vertices. With this characterization, two general constructions of abelian Cayley graphs having fractional revival is presented. Meanwhile, we establish several new families of abelian Cayley graphs admitting fractional reviv…
Perfect State Transfer on Abelian Cayley Graphs
完美态转移（PST）在量子信息处理和量子计算中具有重要的应用价值。本文给出了具有PST的连通简单Cayley图$Γ={\rm Cay}(G,S)$的一个刻画.我们证明了以前关于循环图（其中基础群$G$是循环的）和立方体图（$G=(\mathbb{F}_2^n,+)$）的周期性和PST存在性的许多结果可以从我们的刻画中以统一和更简单的方式导出或推广到任意阿贝尔情况。我们还得到了一些新的结果，包括对以前提出的一些问题的回答。
Perfect state transfer and Cayley presentations
本文研究Cayley图的完美状态转移，搭建经典群同构问题与量子游走的桥梁，证明多类群的Cayley图性质，构造首个无阿贝尔表示的无限族完美状态转移Cayley图及峰值转移图族。
Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups
证明了阿贝尔群上的整Cayley图的距离幂也是该群上的整Cayley图。证明了阿贝尔群上整Cayley图的距离矩阵具有整谱。
Pretty good state transfer in Grover walks on abelian Cayley graphs
本文研究图上Grover游走中的相当好状态转移（PGST），利用Chebyshev多项式推导出图及阿贝尔Cayley图上发生PGST的充要条件，并完整刻画了酉Cayley图上的PGST，给出了无限多族仅有PGST而无完美状态转移的图。
A complete classification of perfect state transfer in Grover walks on abelian Cayley graphs
该研究对有限阿贝尔群上表现出完美状态转移的连通凯莱图进行完全分类，确定其度数为1、2或4，给出度数为4时的充要条件，并将相关循环图分类结果扩展至任意有限阿贝尔群上的所有度数凯莱图。
In this paper, we study self-complementary distance-regular Cayley graphs over abelian groups. We prove that if a regular graph is self-complementary distance-regular, then it is self-complementary strongly regular. We also deal with self-complementary strongly regular Cayley graphs over abelian groups and give an example of a self-complementary strongly regular Cayley graph over a non-elementary abelian group.
In this paper, we consider regular automorphism groups of graphs in the RT$2$ family and the Davis-Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non-abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT$2$ family and Davis-Xiang family and their amorphic abelian Cayley schemes to produce amorphic non-abelian Cayley schemes.
Enumerating typical abelian coverings of Cayley graphs
本文通过将凯莱图的典型阿贝尔覆盖枚举问题归结为有限阿贝尔群子群的枚举，从而完成了相关计数工作。
Fractional revival on Cayley graphs over abelian groups
本文研究有限阿贝尔群上凯莱图的分数复兴问题，给出其存在的充要条件，并应用于循环图与立方体类图的刻画。
Algebraic degrees of $n$-Cayley digraphs over abelian groups
本文研究阿贝尔群上 n-Cayley 有向图，确定其分裂域并给出代数次数的界，进而将结果推广到非阿贝尔群上的 Cayley 有向图。
Fractional revival on semi-Cayley graphs over abelian groups
本文研究了有限阿贝尔群上半凯莱图分数复苏的存在性，给出了充要条件、整数性必要条件及最短时间刻画，并提供了广义二面体群和双循环群上凯莱图的应用实例。
Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified. A nature question is to classify regular Cayley maps of elementary abelian $p$-groups $Z_p^n$. In this paper, a complete classification of regular Cayley maps of $Z_p^n$ is given and moreover, the number of these maps and their genera are enumerated.
In this paper we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let $A$ be an abelian group and let $ι$ be the automorphism of $A$ defined by $a^ι=a^{-1}$, for every $a\in A$. A Cayley graph $\Cay(A, S)$ is said to have an automorphism group as small as possible if $\Aut(\Cay(A,S)) = \langle A,ι\rangle$. In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley di…
A method to determine algebraically integral Cayley digraphs on finite Abelian group
过去的研究者们研究了Cayley有向图或图的特征值。本文主要研究有限阿贝尔群G上的Cayley有向图的特征，其特征值是数域K中的代数整数.通过证明定理1，我们成功地找到了这样做的方法。同时，计算了这类Cayley有向图的个数.
Cubic edge-transitive bi-Cayley graphs over inner-abelian p-groups
一个图称为群H上的双Cayley图，如果它允许H是一个具有两个轨道的半正则作用在其顶点上的自同构自同构群.一个非阿贝尔群称为内阿贝尔群，如果它的所有真子群都是阿贝尔群.本文完成了奇素数p的内交换p-群上连通三次边传递双Cayley图的分类。
On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2
本文将积分Cayley图的刻画从广义二面体群和广义双循环群推广到任意具有指数为2的阿贝尔子群的有限非阿贝尔群，并统一刻画了这类群上的积分混合Cayley图。
Using the domination parameters of Cayley graphs constructed out of $\mathbb{Z}_{p}\times \mathbb{Z}_{m}$, where $m\in\{p^α, p^αq^β, p^αq^βr^γ\},$ in this paper we are discussing about the total and connected domination number and diameter of these Cayley graphs.
Comments on "Extremal Cayley digraphs of finite Abelian groups" [Intercon. Networks 12 (2011), no. 1-2, 125--135]
我们评论的文件“极值凯莱有向图有限阿贝尔群“[Intercon. Networks 1 - 2（2011），no. 1-2，125- 135]。特别地，我们给出了一些反例的结果，并提供了一个正确的结果度2。
Cayley graphs on abelian groups
设$A$是一个阿贝尔群，$ι$是由$i:a\mapsto a^{-1}$定义的$A$的自同构.一个Cayley图$Γ=\mathrm{Cay}(A,S)$称为有一个自同构群\emph{as small as possible}，如果$\mathrm{Aut}(Γ)= A\rtimes\langle i\rangle$ .本文证明了几乎所有阿贝尔群上的Cayley图都有尽可能小的自同构群，证明了Babai和Godsil的一个猜想。
In 1988, Duval introduced the concept of directed strongly regular graphs, which can be viewed as a directed graph version of strongly regular graphs. Such directed graphs have similar structural and algebraic properties to strongly regular graphs. In the past three decades, it was found that Cayley graphs, especially those over dihedral groups, play a key role in the construction of directed strongly regular graphs. In this paper, we focus on the characterization of directed strongly regular Cayley graphs over more general groups. Let $G$ be a non-abelian group with an abelian subgroup of ind…
A geometric method for obtaining an infinite family of Cayley digraphs of constant density on finite Abelian groups is presented. The method works for any given degree and it is based on consecutive dilates of a minimum distance diagram associated with a given initial Cayley digraph. The method is used to obtain infinite families of dense or asymptotically dense Cayley digraphs. In particular, for degree $d=3$, an infinite family of maximum known density is proposed.
A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $Γ$ be an abelian group. The \textit{mixed Cayley graph} $Cay(Γ,S)$ is a mixed graph on the vertex set $Γ$ and edge set $\{ (a,b): b-a\in S \}$, where $0\not\in S$. We characterize integral mixed Cayley graph $Cay(Γ,S)$ over abelian group in terms of its symbol set $S$.
In this paper, we consider the problem on the existence of perfect state transfer(PST for short) on semi-Cayley graphs over abelian groups (which are not necessarily regular), i.e on the graphs having semiregular and abelian subgroups of automorphisms with two orbits of equal size. We stablish a characterization of semi-Cayley graphs over abelian groups having PST. As a result, we give a characterization of Cayley graphs over groups with an abelian subgroup of index 2 having PST, which improves the earlier results on Cayley graphs over abelian groups, dihedral groups and dicyclic group and det…
Cayley compactifications of abelian groups
在Rieffel的基础上，我们定义了具有特定生成集的阿贝尔群的Cayley紧化.我们使用离散几何和交换代数的方法研究它的结构。
On distance-regular Cayley graphs on abelian groups
令$G$表示单位元为1的有限阿贝尔群，令$S$表示$G \setminus {1}$的逆闭子集，它生成$G$，并且存在$s \in S$，使得$\la S \setminus \{s,s^{-1}\} \ra \ne G$。本文给出了距离正则Cayley图$\cay(G;S)$对$G$和$S$的完全分类.
A graph $Γ$ is said to be a semi-Cayley graph over a group $G$ if it admits $G$ as a semiregular automorphism group with two orbits of equal size. We say that $Γ$ is normal if $G$ is a normal subgroup of ${\rm Aut}(Γ)$. We prove that every connected intransitive one-matching semi-Cayley graph, with maximum degree three, over a finite abelian group is normal and characterize all such non-normal graphs.
Abelian Cayley digraphs with asymptotically large order for any given degree
阿贝尔Cayley有向图可以通过将$Z$中的同余概念推广到$Z^n$来构造。在这里，我们使用这种方法来提出一个家庭这样的有向图，其中，对于每一个固定值的程度，有渐近大量的顶点直径增加。到目前为止，最著名的渐近稠密的结果都是非建设性的。
Classes of Symmetric Cayley Graphs over Finite Abelian Groups of Degrees 4 and 6
本文研究了有限阿贝尔群上度为4和6的对称无向Cayley图族的特征.
Fractional revival on quasi-abelian Cayley graphs
针对量子自旋网络中的分数复兴问题，本文刻画连接集为共轭类并集的拟阿贝尔 Cayley 图，给出其发生分数复兴的充要条件，并推广已有阿贝尔群结果。
Strongly regular Cayley graphs over primary abelian groups of rank 2
秩为2的阿贝尔群上具有Paley参数的强正则Cayley图在[J. ADavis，PartialDifferenceSetsinP-groups，Arch.Math.63（1994）103-110; K. HLeung，S.L. Ma，Partial difference sets with Paley parameters，Bull. London Math Soc.27（1995）553-564].证明了这类图存在当且仅当对应的群同构于${\mathbb Z}_{p^n} \oplus {\mathbb Z}_{p^n}$，其中$p$是奇素数.本文利用Schur环方法对这类群上的强正则Cayley图进行了分类。作为结果，我们得到了秩为2的阿贝尔群上具有Paley参数的强正则Cayley图的完全分类。
We construct a polynomial-time algorithm which given a graph $Γ$ finds the full set of non-equivalent Cayley representations of $Γ$ over the group $D\cong C_p\times C_{p^k}$, where $p\in\{2,3\}$ and $k\geq 1$. This result implies that the recognition and the isomorphism problems for Cayley graphs over $D$ can be solved in polynomial time.
Which finitely generated Abelian groups admit isomorphic Cayley graphs?
我们证明了非正则生成的阿贝尔群的Cayley图是刚性的.作为结果，我们得到两个生成的阿贝尔群允许同构Cayley图当且仅当它们具有相同的秩和它们的挠部分具有相同的基数。证明只使用基本参数，并制定了几何语言。
Fractional revival on oriented Cayley and semi-Cayley graphs over abelian groups
本文研究定向图上的真分数复兴存在性，建立定向图强共谱顶点间真FR的充要条件，证明阿贝尔群上定向Cayley图无真FR，并刻画阿贝尔群上定向半-Cayley图的真FR存在条件。
We study the game of Cops and Robbers, where cops try to capture a robber on the vertices of a graph. Meyniel's conjecture states that for every connected graph $G$ on $n$ vertices, the cop number of $G$ is upper bounded by $O(\sqrt{n})$, i.e., that $O(\sqrt{n})$ suffice to catch the robber. We present several families of abelian Cayley graphs that are Meyniel extremal, i.e., graphs whose cop number is $O(\sqrt{n})$. This proves that the $O(\sqrt{n})$ upper bound for Cayley graphs proved by Bradshaw is tight up to a multiplicative constant. In particular, this shows that Meyniel's conjecture, …
Perfect codes in quintic Cayley graphs on abelian groups
本文定义了图中的完美码，并分类了阿贝尔群上所有存在完美码的连通五次凯莱图，同时完全确定了此类图的所有完美码。
We show that for certain integers $n$, the problem of whether or not a Cayley digraph $Γ$ of $\mathbb Z_n$ is also isomorphic to a Cayley digraph of some other abelian group $G$ of order $n$ reduces to the question of whether or not a natural subgroup of the full automorphism group contains more than one regular abelian group up to isomorphism (as opposed to the full automorphism group). A necessary and sufficient condition is then given for such circulants to be isomorphic to Cayley digraphs of more than one abelian group, and an easy-to-check necessary condition is provided.
Perfect state transfer on bi-Cayley graphs over abelian groups
本文针对任意有限阿贝尔群上的双凯莱图，建立了完美状态转移的充分必要条件，并由此统一导出多类群上凯莱图的已知与新结果，还构造了二面体群上连通非正规凯莱图存在完美状态转移的反例。
A mixed graph is called \emph{second kind hermitian integral}(or \emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. Let $Γ$ be an abelian group. We characterize the set $S$ for which a mixed Cayley graph $\text{Cay}(Γ, S)$ is HS-integral. We also show that a mixed Cayley graph is Eisenstein integral if and only if it is HS-integral.
Integral Cayley multigraphs over Abelian and Hamiltonian groups
本文证明了群$S$上的Cayley多重图是整数图（即，如果$S$位于由$G$的正规子群生成的布尔代数上的积分锥中，则它的所有特征值都是整数。当$G$是阿贝尔时，反之亦然。这特别给出了Bridges和Mena（1982）定理的另一种性质理论证明。我们推广了这一结果，给出了Hamilton群上的Cayley多重图是整图的一个充要条件.
A well-known conjecture of Alspach says that every $2k$-regular Cayley graph of an abelian group can be decomposed into Hamiltonian cycles. We consider an analogous question for infinite abelian groups. In this setting one natural analogue of a Hamiltonian cycle is a spanning double-ray. However, a naive generalisation of Alspach's conjecture fails to hold in this setting due to the existence of $2k$-regular Cayley graphs with finite cuts $F$ where $|F|$ and $k$ differ in parity, which necessarily preclude the existence of a decomposition into spanning double-rays. We show that every $4$-regul…
Recognizing and testing isomorphism of Cayley graphs over an abelian group of order $4p$ in polynomial time
我们构造了一个多项式时间算法，给定一个图$X$与$4p$顶点（$p$是素数），发现（如果有的话）的凯莱表示的$X$在组$C_2\times C_2\times C_p$。这一结果与已知的循环图的类似结果一起表明，在$4p$阶的阿贝尔群上的Cayley图的同构的识别和检验可以在多项式时间内完成.
Perfect codes in 2-valent Cayley digraphs on abelian groups
研究阿贝尔群上强连通2价Cayley有向图的完美码问题，完成图的分类并完全确定其所有完美码。
Perfect state transfer in Grover walks on normal Cayley graphs
研究正规凯莱图上格罗弗游走的完美状态转移，建立充要条件，推导阿贝尔群等上凯莱图的谱准则及组合特征，给出酉凯莱图完美状态转移完整特征，证明其中恰有四个图展示该转移。
Cops and robbers on directed and undirected abelian Cayley graphs
本文通过精细归纳法证明阿贝尔群上有向与无向凯莱图的警察数上界为$O(\sqrt{n})$，改进无向情形并首次建立有向情形上界，同时构造了警察数为$Θ(\sqrt{n})$的Meyniel极值族。
Asymptotic stability of Cayley graphs on abelian groups
本文证明几乎所有有限阿贝尔群的赛雷图都是稳定的，通过研究其常规双覆盖的自同构群最小化问题。
Isomorphisms of abelian Cayley graphs with their natural edge-colouring
本文证明阿贝尔群连通凯莱图保持自然边着色的同构可分解为群同构与保色图自同构的复合，并给出保色置换自同构均为仿射映射的充要条件。
A formula for eigenvalues of integral Cayley graphs over abelian groups
本文针对阿贝尔群上连接集为 $[x]$ 的整凯莱图，给出了其特征值的代数公式，该公式涉及莫比乌斯函数的类似物。
Asymptotic metric behavior of random Cayley graphs of finite abelian groups
利用Marklof和Strömbergsson的方法，我们建立了有限阿贝尔群的随机Cayley图关于随机选取的一组固定大小的生成元的度量参数的几个极限律.这样做，我们解决了阿米尔和古雷尔古列维奇的猜想。
Fourier analysis on distance-regular Cayley graphs over abelian groups
针对任意群类上距离正则Cayley图的刻画问题，本文利用阿贝尔群上的傅里叶分析建立其与有限几何组合对象的联系，结合经典结果分类了群$\mathbb{Z}_n \oplus \mathbb{Z}_p$（$p$为奇素数）上的所有距离正则Cayley图。
Let $X$ be a connected Cayley graph on an abelian group of odd order, such that no two distinct vertices of $X$ have exactly the same neighbours. We show that the direct product $X \times K_2$ (also called the "canonical double cover" of $X$) has only the obvious automorphisms (namely, the ones that come from automorphisms of its factors $X$ and $K_2$). This means that $X$ is "stable". The proof is short and elementary. The theory of direct products implies that $K_2$ can be replaced with members of a much more general family of connected graphs.
We show that any connected Cayley graph $Γ$ on an Abelian group of order $2n$ and degree $\tildeΩ(\log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $Γ$ is bipartite. Our proof is based on Sapozhenko's graph container method and uses the Plünnecke-Rusza-Petridis inequality from additive combinatorics.
On the k-spanning cyclability of 4-valent Cayley graphs on Abelian groups
本文研究了阿贝尔群上4-valent Cayley图的$k$-spanning cyclability，即对任意$k$个不同顶点，存在由$k$个环组成的2-factor使每个顶点属于不同环。
Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case in which such graphs are Cayley graphs of Abelian groups. These groups can be constructed by using a generalization to $\mathbb{Z}^n$ of the concept of congruence in $\mathbb{Z}$. Here we use this approach to present some families of mixed graphs, which, for every fixed value of the degree, have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.
Algebraic degrees of quasi-abelian semi-Cayley digraphs
本文利用群的不可约特征标，确定了任意有限群上拟阿贝尔半凯莱有向图的分裂域和代数次数，推广了此前关于凯莱图与半凯莱有向图代数次数的相关研究。
Fast robbers on abelian Cayley graphs and digraphs
该研究针对有限强连通阿贝尔凯莱有向图的警察与强盗游戏快速强盗版本，推导了有界出度下的复杂度上界及缓增度区域的一致次线性界，改进了无向情形的结果。
The splitting field $\mathbb{SF}(Γ)$ of a mixed graph $Γ$ is the smallest field extension of $\mathbb{Q}$ which contains all eigenvalues of the Hermitian adjacency matrix of $Γ$. The extension degree $[\mathbb{SF}(Γ):\mathbb{Q}]$ is called the algebraic degree of $Γ$. In this paper, we determine the splitting fields and algebraic degrees of mixed Cayley graphs over abelian groups. This generalizes the main results of [K. Mönius, Splitting fields of spectra of circulant graphs, J. Algebra 594(15) (2022) 154--169] and [M. Kadyan, B. Bhattacharjya, Integral mixed Cayley graphs over abelian groups…
On automorphisms and structural properties of generalized Cayley graphs
本文研究了广义Cayley图。证明了任意2 p阶广义Cayley图都是Cayley图，其中p是素数.特别关注的是阿贝尔群上的广义Cayley图。证明了阿贝尔群上关于作为逆的自同构的广义Cayley图是Cayley图当且仅当该群是初等阿贝尔2-群或其Sylow 2-子群是循环的.给出了广义Cayley图为无价值图的充要条件。
Cayley graphs of abelian groups, circular colorings and discrete fundamental groups
该研究针对每个整数k≥2，刻画了满足特定指数条件的阿贝尔群，证明其与不存在圆色数小于2+2/(k-1)的非二部Cayley图等价，改进了相关代数刻画。
On CI-property of normal Cayley digraphs over abelian groups
本文研究阿贝尔群上正规 Cayley 有向图的 CI-性质，将任意阿贝尔群情形归约为阿贝尔 p-群，并证明了阶数不超过 $p^5$ 的阿贝尔 p-群上的正规 Cayley 有向图均为 CI 的。
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