Joy Morris
A circulant (di)graph is a (di)graph on n vertices that admits a cyclic automorphism of order n. This paper provides a survey of the work that has been done on finding the automorphism groups of circulant (di)graphs, including the generalisation in which the edges of the (di)graph have been assigned colours that are invariant under the aforementioned cyclic automorphism.
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We show that almost all circulant graphs have automorphism groups as small as possible. Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost all circulant graphs whose order lies in any one of these families, are normal. That almost all Cayley (di)graphs whose automorphism group is not as small as possible are normal was conjectured by the second author, so these results provide counterexamples to this conjecture. It is then shown that there is a large family of integers for which almost every c…
The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups.
Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an association scheme on triples (AST) and constructed examples of several families of ASTs. Many of their examples used 2-transitive permutation groups: the non-trivial ternary relations of the ASTs were sets of ordered triples of pairwise distinct points of the underlying set left invariant by the group; and…
In this paper, we begin by partitioning the edges (or arcs) of a circulant (di)graph according to which generator in the connection set leads to each edge. We then further refine the partition by subdividing any part that corresponds to an element of order less than $n$, according to which of the cycles generated by that element the edge is in. It is known that if the (di)graph is connected and has no multiple edges, then any automorphism that respects the first partition and fixes the vertex corresponding to the group identity must be an automorphism of the group (this is in fact true in the …
A graph $X$ is said to be unstable if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if it is unstable, connected, and non-bipartite, and no two distinct vertices of X have exactly the same neighbors. We find all of the nontrivially unstable circulant graphs of valency at most $7$. (They come in several infinite families.) We also show that the instability of each of these graphs is explained by theorems of Steve Wilson. This is best possible, becau…
A graph $X$ is said to be "unstable" if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is "nontrivially unstable" if it is unstable, connected, and nonbipartite, and no two distinct vertices of X have exactly the same neighbors. We find three new conditions that each imply a circulant graph is unstable. (These yield infinite families of nontrivially unstable circulant graphs that were not previously known.) We also find all of the nontrivially unstable circulant graphs of o…
Twins and Co-Twins in Circulant graphs
本文研究环形图中双胞胎和共双胞胎的结构，通过简化自 automorphism 组的分析，揭示了顶点传递图的对称性参数。
For an integer $k\geq 1$, a graph is called a $k$-circulant if its automorphism group contains a cyclic semiregular subgroup with $k$ orbits on the vertices. We show that, if $k$ is even, there exist infinitely many cubic arc-transitive $k$-circulants. We conjecture that, if $k$ is odd, then a cubic arc-transitive $k$-circulant has order at most $6k^2$. Our main result is a proof of this conjecture when $k$ is squarefree and coprime to $6$.
Automorphism groups of endomorphism monoids of $0$-unit $3$-valent circulant digraphs
本文研究 $0$-单位 $3$-度循环有向图的自同态幺半群的自同构群，通过建立无 $2$-圈单位循环有向图存在诱导有向圈的自同态的充要条件，显式确定了正规化子群 $U_S(\mathbb{Z}_n)$，解决了该族的未决问题。
The distinguishing chromatic number of a graph $G$ is the smallest number of colors needed to properly color the vertices of $G$ so that the trivial automorphism is the only symmetry of $G$ that preserves the coloring. We investigate the distinguishing chromatic number for Hamiltonian circulant graphs with maximum degree at most 4.
Symmetry Parameters of Two-Generator Circulant Graphs
本文研究由单顶点双环电压图导出的双生成元循环图，完整刻画其自同构群，并确定其确定数、区分数及2-区分代价，同时将结果推广到相应的细分图。
Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ is the least integer $r$ such that there is a $r$-labeling of the vertices of $G$ that is not preserved by any nontrivial automorphism of $G$. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs depending on $n$. In this paper, we study circulant graphs of order $n$ where the adjacency is defined using a symmetric subset $A$ of $\mathbb{Z}_n$, called generator. We give a construction of a family of…
New classes of groups related to algebraic combinatorics with applications to isomorphism problems
本文引入$5/2$-群类的两个精化，证明其正则循环子群共轭，从而推广并简化了对称构型与单位循环有向图同构的判定证明。
On the automorphism group of direct product of digraphs
研究有向图直积自同构群等于因子自同构群直积的条件，给出连通有向图不稳定的充要条件，并应用于循环有向图。
Spectra and the exact number of Automorphisms of Paley-type graphs of order a product of n distinct primes
该研究将Paley型图的研究从两不同毕达哥拉斯素数乘积阶拓展至n个不同毕达哥拉斯素数乘积阶，确定了这类图的邻接谱与自同构精确数目。
On distance magic circulants of valency 6
本文研究价为6的距离魔幻循环图，给出其成为距离魔幻图的若干必要与充分条件，并完整分类阶不被12整除的情形。
On cubic polycirculant nut graphs
本文研究三次ℓ-循环核图的存在性，通过计算机辅助证明排除ℓ=4和5的情形，并构造性证明当ℓ∈{6,7}或ℓ≥9时存在无穷多个此类图。
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.
Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks
本文完全刻画了度不超过4的循环图中Grover游走具有完美态转移的图，发现度为3的图均不具备，度为4的图仅存在于两个无限族中，并利用图的对称性与特征值及分圆域整数环结果推导了必要条件。
We consider circulant graphs having $p$ vertices, with $p$ prime. To any such graph we associate a certain number $k$, that we call type of the graph. We prove that for $p>>k$ the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.
A Frobenius group is a transitive permutation group which is not regular but only the identity element can fix two points. Such a group can be expressed as the semi-direct product $G = K \rtimes H$ of a nilpotent normal subgroup $K$ and another group $H$ fixing a point. A first-kind $G$-Frobenius graph is a connected Cayley graph on $K$ with connection set an $H$-orbit $a^H$ on $K$ that generates $K$, where $H$ has an even order or $a$ is an involution. It is known that the first-kind Frobenius graphs admit attractive routing and gossiping algorithms. A complete rotation in a Cayley graph on a…
Automorphism Groups and Structure of 4-Valent Cayley Graphs on Dihedral Groups
研究二面体群上4度Cayley图的自同构群与结构，通过分析不同生成集的Cayley图性质，确定其分类与构造方法。
Two methods, structural (constructive) and multiplier (analytical), of exact enumeration of undirected and directed circulant graphs of orders 27 and 125 are elaborated and represented in detail here together with intermediate and final numerical data. The first method is based on the known useful classification of circulant graphs in terms of $S$-rings and results in exhaustive listing (with the use of COCO and GAP) of all corresponding $S$-rings of the indicated orders. The latter method is conducted in the framework of a general approach developed earlier for counting circulant graphs of pr…
Cyclic relative difference sets and circulant weighing matrices
本文扩展了对循环相对差集的搜索，并将其应用于循环权矩阵的存在性研究。
Towards inductive proofs in algebraic combinatorics
本文引入了一类新的传递置换群，给出了其商保持在该族中的充分条件，并利用归纳法确定了阶为$p^n$的循环有向图的自同构群。
We survey recent developments concerning two properties of classes of finite structures: the Ramsey property and the extension property for partial automorphisms (EPPA).
On the existence of directed strongly regular graphs with parameters (22, 9, 6, 3, 4)
本文构造了一族参数为 (22, 9, 6, 3, 4) 的有向强正则图，其邻接矩阵由 3×3 循环块构成，自同构群为 Z_3，并用骨架与索具概念刻画其结构。
A symmetric conference matrix of order 86
本文构造了86阶对称会议矩阵，解决了该最小未解阶数的存在性问题，并等价得到会议图、正则双图及等角紧框架，其86个后代产生十个非同构强正则图。
In this paper we survey existing results on Deza graphs and give some new results. We present an introduction to Deza graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of Deza graphs since the initial paper by five authors [M. Erickson, S. Fernando, W. H. Haemers, D. Hardy, J. Hemmeter, Deza graphs: \emph{A generalization of strongly regular graphs}, J. Comb. Designs. 7 (1999), 395--405.] was written. We then investigate 3-class cyclotomic schemes and give necessary and sufficient conditions to get a Deza graph as a graph given…
Ramsey number $R(4, 20) \ge 252$
该研究构造了两个251阶显式循环图，证明R(4,20)≥252，改进了拉姆齐数的相关下界，为拉姆齐数研究提供了新的构造方法。
On an infinite sequence of strongly regular digraphs with parameters $(9(2n+3), 3(2n+3), 2n+4, 2n+1, 2n+4)$
本文构建了无限序列的强正则有向图，通过将邻接矩阵表示为由循环块组成的块矩阵，并结合多项式模$x^{2n+3}-1$的紧缩操作，发现稳定的结构模式，从而推导出邻接矩阵的显式公式，并给出了两个此前未解决的图参数例子。
Let C be a binary linear code and suppose that its automorphism group contains a non trivial subgroup G. What can we say about C knowing G? In this paper we collect some answers to this question in the cases G=C_p, G=C_2p and G=D_2p (p an odd prime), with a particular regard to the case in which C is self-dual. Furthermore we generalize some methods used in other papers on this subject. Finally we give a short survey on the problem of determining the automorphism group of a putative self-dual [72,36,16] code, in order to show where these methods can be applied.
This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks and comments.
A survey on constructive methods for the Oberwolfach problem and its variants
本文综述了Oberwolfach问题及其变体的构造性方法，重点介绍放大构造、二部因子情形、循环图方法及具有良好自同构性质的解。
Generating the Spanning Trees of Series-Parallel Graphs up to Graph Automorphism
本文针对串联-并联图，提出在图形自同构意义下生成非等价生成树的输出线性时间算法，覆盖定向、半定向和无定向三种情形，并揭示自同构群与生成树递归结构的对应关系。
All generalized rose window graphs are hamiltonian
本文针对连通双循环图的哈密顿性提出猜想，并证明该猜想对所有四次、s=2的广义玫瑰窗图成立。
We survey properties of the set of possible exponents of subsets of $\Z_n$ (equivalently, exponents of primitive circulant digraphs on $n$ vertices). Let $E_n$ denote this exponent set. We point out that $E_n$ contains the positive integers up to $\sqrt{n}$, the `large' exponents $\lfloor \frac{n}{3} \rfloor +1, \lfloor \frac{n}{2} \rfloor, n-1$, and for even $n \ge 4$, the additional value $\frac{n}{2}-1$. It is easy to see that no exponent in $[\frac{n}{2}+1,n-2]$ is possible, and Wang and Meng have shown that no exponent in $[\lfloor \frac{n}{3}\rfloor +2,\frac{n}{2}-2]$ is possible. Extend…
Symmetries of the Honeycomb toroidal graphs
本文针对蜂窝环面图这一参数化三正则 Cayley 图族，确定了每个图的完整自同构群，从而解决 Alspach 综述中提出的两个研究问题。
Computing equivalence classes of finite group actions on orientable surfaces: A dynamic survey
本文研究黎曼曲面上有限群作用的拓扑等价分类问题，通过富克斯群自同构作用分析并借助计算机代数系统，给出了亏格 g≤9 的完整分类列表。
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。
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