Grigory Ryabov
A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_4\times C_p^2$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$.
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In this paper we study finite groups which have Cayley isomorphism property with respect to Cayley maps, CIM-groups for a brief. We show that the structure of the CIM-groups is very restricted. It is described in Theorem~\ref{111015a} where a short list of possible candidates for CIM-groups is given. Theorem~\ref{111015c} provides concrete examples of infinite series of CIM-groups.
On the Isomorphism Problem of Cayley Graphs of Graph Products
该研究探讨图乘积的Cayley图同构问题，证明顶点群的Cayley图同构可推出对应图乘积的Cayley图同构，为相关同构判定提供理论依据。
Two finitely generated monoids are constructed, one finitely presented the other not, whose (directed, unlabelled) Cayley graphs are isomorphic.
For every prime $p>3$ we prove that $Q \times \mathbb{Z}_p$ and $\mathbb{Z}_2^3 \times \mathbb{Z}_p$ are DCI- groups. This result completes the description of CI-groups of order $8p$.
For every prime $p > 3$ and for every prime $q>p^3$ we prove that $\mathbb{Z}_q \times \mathbb{Z}_p^3$ is a DCI-group.
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce Cayley tournaments, which are special Cayley digraphs, and show that there is a one to one identification between the isomorphism classes of vertex-transitive tournaments with prime order and the isomorphism classes of Cayley tournaments with prime order.
A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly(n).
Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G;S) and Cay(G';S') are connected Cayley graphs of finite valency on two nilpotent groups G and G', then every isomorphism from Cay(G;S) to Cay(G';S') factors through to a well-defined affine map from G/N to G'/N', where N and N' are the torsion subgroups of G and G', respectively. F…
A finite group $G$ is called a DCI-group if two Cayley digraphs over $G$ are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group $C_2^5\times C_p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$. Together with the previously obtained results, this implies that a group $G$ of order $32p$, where $p$ is a prime, is a DCI-group if and only if $p\neq 2$ and $G\cong C_2^5\times C_p$.
The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel results. Some interesting vertex-transitive graphs are given, including a first example of connected symmetric non-Cayley non-GI-graph. Also, we initiate the study for GI and DGI-groups, defined analogously to the concept of CI and DCI-groups.
On isomorphisms of $m$-Cayley digraphs
本文研究m-Cayley有向图的同构问题，通过刻画自同构群中G的正规化子并推广CI性质，定义了mCI、mPCI、mDCI和mPDCI等概念，给出Babai型判据，并分类了m≥2时的有限mDCI群和m≥4时的有限mPDCI群。
We construct a polynomial-time algorithm that given a graph $X$ with $4p$ vertices ($p$ is prime), finds (if any) a Cayley representation of $X$ over the group $C_2\times C_2\times C_p$. This result, together with the known similar result for circulant graphs, shows that recognising and testing isomorphism of Cayley graphs over an abelian group of order $4p$ can be done in polynomial time.
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。
Isomorphism factorizations of the complete graph into Cayley graphs on CI-groups
本文研究了将完全图分解为CI群上的Cayley图的等价因子分解问题，提出了相关条件和构造方法。
Let $p$ be an odd prime, and $D_{2p}=\langle τ,σ\mid τ^p=σ^2=e,στσ=τ^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we provide the number of (connected) Cayley (di-)graphs on $D_{2p}$ up to isomorphism by using the Pólya enumeration theorem. In the process, we also enumerate (connected) Cayley digraphs on $D_{2p}$ of out-degree $k$ up to isomorphism for each $k$.
On kernel isomorphisms of $m$-Cayley digraphs and finite $2$PCI-groups
本文研究$m$-Cayley有向图同构问题，引入核同构并建立相关理论，证明有限$2$PCI-群的可解性与Sylow子群结构，完成有限非阿贝尔BCI-群的完全分类。
On the BCI Problem
本文推广了群 Haar 图的 BCI 同构问题，提出理论解法，并针对阿贝尔群和奇数阶循环群等特殊情况，将问题归约或完全解决了同构判定。
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.
Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. By a Cayley graph $Cay(G,S)$ we mean the graph whose vertex set is the set of elements of $G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. A group $G$ is called a CI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $S^α=T$ for some automorphism $α$ of $G$. A finite group $G$ is called a BI-group if $Cay(G,S)\cong Cay(G,T)$ for some inverse closed subsets $S$ and $T$ of $G\setminus \{1\}$, then $M_ν^S=M_ν^T$ for all positive integers $ν$, where $M_ν…
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。
Isomorphisms of bi-Cayley graphs on generalized quaternion groups
该文研究广义四元数群上双Cayley图的同构问题，刻画了m=2、3时阶4n(n≥2)的广义四元数群成为m-BCI-群的等价条件，关联了m-BCI、m-DCI、m-CI群的判定。
Composing group automorphisms with colour-preserving automorphisms of Cayley graphs
本文证明连通有限Cayley图的保色置换自同构在阶为奇数或无平方因子数时可分解为群自同构与保色图自同构的复合，并推广到不同图之间的同构。
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.
On Isomorphisms of Tetravalent Cayley Digraphs over Dihedral Groups
本文研究二面体群上四度Cayley有向图的同构，给出了4-DCI-群和4-CI-群的充要条件，即n为奇数且不被9整除时是4-DCI-群，n为奇数时是4-CI-群。
We explicitly determine all of the transitive groups of degree p-squared, p a prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of order p. Furthermore, we provide a general description of the transitive groups of degree p-squared whose Sylow p-subgroup is such a wreath product, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p-squared, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p-squared, and find all nonnormal Cayley grap…
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $α\in Aut(G)$ such that $S^α=T$, whenever $Cay(G,S)\cong Cay(G,T)$ for a Cayley (di)graph $Cay(G,T)$. A finite group $G$ is called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively. Motivated by a conjecture …
Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on Sn with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of Sn. In this paper, we prove three results on normal Cayley graphs on Sn possessing this property fo…
Finding automorphism groups of double coset graphs and Cayley graphs are equivalent
本文证明凯莱图与双陪集图的自同构群可互相求得，并在已知所有相关群的条件下证明二者的同构问题等价，主要工具是基于连接集识别凯莱图是否为圈积的识别定理。
Let $p$ be an odd prime, and $D_{2p}=\langle a,b\mid a^p=b^2=1,bab=a^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we completely classify the cubic Cayley graphs on $D_{2p}$ up to isomorphism by means of spectral method. By the way, we show that two cubic Cayley graphs on $D_{2p}$ are isomorphic if and only if they are cospectral. Moreover, we obtain the number of isomorphic classes of cubic Cayley graphs on $D_{2p}$ by using Gauss' celebrated law of quadratic reciprocity.
Enumerating Cayley digraphs on dihedral groups
该研究聚焦二面体群上Cayley有向图的CI-同构计数，借助Cauchy-Frobenius引理与自同构性质，推导了具有DCI性质的二面体群（特别是$\text{D}_{6p}$，p为大于3的素数）上非同构Cayley有向图数的显式公式。
Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half its vertices has maximum degree at least $\sqrt{d}$, where $d$ is the valency of the hypercube. This was generalised by Alon and Zheng who proved that every Cayley graph on an elementary abelian $2$-group has the same property. Very recently, Potechin and Tsang proved an analogous results for Cayley graphs on abelian groups. They also conjectured that all Cayley graphs have the analogous property. We disprove thi…
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if for every $T\subseteq G$, $Cay(G,S)\cong Cay(G,T)$ implies that there is $σ\in Aut(G)$ such that $S^σ=T$. We call a group $G$ a NDCI-group if all normal Cayley digraphs of $G$ are CI-digraphs, and a NCI-group if all normal Cayley graphs of $G$ are CI-graphs, respectively. In this paper, we prove that a cyclic group of order $n$ is a NDCI-group if and only if…
Lattice polytopes which possess the integer decomposition property (IDP for short) turn up in many fields of mathematics. It is known that if the Cayley sum of lattice polytopes possesses IDP, then so does their Minkowski sum. In this paper, the Cayley sum of the order polytope of a finite poset and the stable set polytope of a finite simple graph is studied. We show that the Cayley sum of an order polytope and the stable set polytope of a perfect graph possesses a regular unimodular triangulation and IDP, and hence so does their Minkowski sum. Moreover, it turns out that, for an order polytop…
The real monomial representations of Clifford algebras give rise to two sequences of bent functions. For each of these sequences, the corresponding Cayley graphs are strongly regular graphs, and the corresponding sequences of strongly regular graph parameters coincide. Even so, the corresponding graphs in the two sequences are not isomorphic, except in the first 3 cases. The proof of this non-isomorphism is a simple consequence of a theorem of Radon.
Let $M$ be a matroid satisfying a matroidal analogue of the Cayley-Bacharach condition. Given a number $k \ge 2$, we show that there is no nontrivial bound on ranks of a $k$-tuple of flats covering the underlying set of $M$. This addresses a question of Levinson-Ullery motivated by earlier results which show that bounding the number of points satisfying the Cayley-Bacharach condition forces them to lie on low-dimensional linear subspaces. We also explore the general question what matroids satisfy the matroidal Cayley-Bacharach condition of a given degree and its relation to the geometry of gen…
Cayley sums and Minkowski sums of lattice polytopes
本文以 Minkowski 和刻画 Cayley 和的整数分解性质，并据此研究 2-凸正规格多面体两类和的该性质，同时讨论其 level 性质。
New criteria for which Cayley graphs of cyclic groups of any order can be completely determined--up to isomorphism--by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley graphs of cyclic groups with the same list of eigenvalues of their adjacency matrices will be presented.
Regular balanced Cayley maps on ${\rm PSL}(2,p)$
本文针对每个素数 p>3，完整分类了群 PSL(2,p) 上的正则平衡凯莱地图，填补了此前仅对循环群、二面体群等特殊群类进行分类的空白。
The Aldous property for normal Cayley graphs on symmetric groups
本文研究对称群上正规Cayley图的Aldous性质，证明了其与补图的性质互斥性，确定了大部分具有该性质的正规Cayley图，解决了2023年提出的开放问题，还完成了线图及特征值相关分类。
In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group $R$ has a nontrivial proper n…
Let $S_n$ denote the symmetric group of degree $n$ with $n\geq 3$. Set $S=\{c_n=(1\ 2\ldots \ n),c_n^{-1},(1\ 2)\}$. Let $Γ_n=\mathrm{Cay}(S_n,S)$ be the Cayley graph on $S_n$ with respect to $S$. In this paper, we show that $Γ_n$ ($n\geq 13$) is a normal Cayley graph, and that the full automorphism group of $Γ_n$ is equal to $\mathrm{Aut}(Γ_n)=R(S_n)\rtimes \langle\mathrm{Inn}(ϕ)\rangle\cong S_n\rtimes \mathbb{Z}_2$, where $R(S_n)$ is the right regular representation of $S_n$, $ϕ=(1\ 2)(3\ n)(4\ n-1)(5\ n-2)\cdots$ $(\in S_n)$, and $\mathrm{Inn}(ϕ)$ is the inner isomorphism of $S_n$ induced b…
Enumeration of Cayley graphs over a nonabelian group of order $8p$
针对8p阶非阿贝尔群$T_{8p}$，利用波利亚计数定理给出互不同构Cayley图及连通Cayley图的计数公式，并列出3≤p≤13时的精确数量。
We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such that every such automorphism of every connected Cayley graph on G has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.
For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containin…
A Cayley graph Cay$(G;S)$ has the CI (Cayley Isomorphism) property if for every isomorphic graph Cay$(G;T)$, there is a group automorphism $α$ of $G$ such that $S^α=T$. The DCI (Directed Cayley Isomorphism) property is defined analogously on digraphs. A group $G$ is a CI-group if every Cayley graph on $G$ has the CI property, and is a DCI-group if every Cayley digraph on $G$ has the DCI property. Since a graph is a special type of digraph, this means that every DCI-group is a CI-group, and if a group is not a CI-group then it is not a DCI-group, but there are well-known examples of groups that…
Subperiodic groups and bounded automorphisms of periodic graphs
研究亚周期群在三维空间中的分类，通过开发易计算的不变量识别同构类，并探讨有限阶有界自同构在晶格图中的应用。
Neumaier Cayley graphs
本文研究Neumaier Cayley图，给出其成为强正则Neumaier Cayley图的充要条件，并刻画了度数不超过10的小度数Neumaier Cayley图的特征。
The spined cube $SQ_n$ is a variant of the hypercube $Q_n$, introduced by Zhou et al. in [Information Processing Letters 111 (2011) 561-567] as an interconnection network for parallel computing. A graph $\G$ is an $m$-Cayley graph if its automorphism group $\Aut(\G)$ has a semiregular subgroup acting on the vertex set with $m$ orbits, and is a Caley graph if it is a 1-Cayley graph. It is well-known that $Q_n$ is a Cayley graph of an elementary abelian 2-group $\mz_2^n$ of order $2^n$. In this paper, we prove that $SQ_n$ is a 4-Cayley graph of $\mz_2^{n-2}$ when $n\geq6$, and is a $\lfloor n/2\…
Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X \subseteq \mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ by another surface $Y \subseteq \mathbb{S}^2$. We also prove that if a group $H$ has a finitely generated Cayley (multi-)graph $C$ covariantly embeddable in $\mathbb{S}^2$, then $C$ can be chosen so as to have no infinite path on the boundary of a face. The proofs of these facts are intertwined, and the classes of groups they define coincide. In t…
In this paper, we show the existence of a polynomial time graph isomorphism algorithm for all graphs excluding graphs that are locally trianglefree. This particular class of graphs allows to divide the graph into neighbourhood sub-graph where each of induced sub-graph (neighbourhood) has at least 2 vertices. We construct all possible permutations for each induced sub-graph using a search tree. We construct automorphisms of subgraphs based on these permutations. Finally, we decide isomorphism through automorphisms . The author expects that the solution, present in this paper, may lead to a fast…
Li and Praeger classified finite nonabelian simple groups, it has only one or two fusion classes of any certain value. As a by-product, they classified m-CI-groups, which is critical in the research of Cayley graphs. In the paper, we will consider generalized Cayley graphs. This concept is proposed by Marusic et al. In the paper, (local) m- GCI-group is defined, and we get many properties and characterizations based on the generalized Cayley isomorphism, which are the key measures for the classification of (local) m-GCI-group. And above all, we will give a classification of local 2-GCI-groups …
A finite group R is a CI-group if, whenever S and T are subsets of R with the Cayley graphs Cay(R,S) and Cay(R,T) isomorphic, there exists an automorphism x of R with S^x=T. The classification of CI-groups is an open problem in the theory of Cayley graphs and is closely related to the isomorphism problem for graphs. This paper is a contribution towards this classification, as we show that every dihedral group of order 6p, with p>3 prime, is a CI-group.
Given a finite Abelian group $G$ and a generator subset $A\subset G$ of cardinality two, we consider the Cayley digraph $Γ=$Cay$(G,A)$. This digraph is called $2$--Cayley digraph. An extension of $Γ$ is a $2$--Cayley digraph, $Γ'=$Cay$(G',A)$ with $G<G'$, such that there is some subgroup $H<G'$ satisfying the digraph isomorphism Cay$(G'/H,A)\cong$Cay$(G,A)$. We also call the digraph $Γ$ a quotient of $Γ'$. Notice that the generator set does not change. A $2$--Cayley digraph is called optimal when its diameter is optimal with respect to its order. In this work we define two procedures, E and Q,…
Two colourings of a graph are orthogonal if they have the property that when two vertices are coloured with the same colour in one colouring, then those vertices receive distinct colours in the other colouring. In this paper, orthogonal colourings of Cayley graphs are discussed. Firstly, the orthogonal chromatic number of cycle graphs are completely determined. Secondly, the orthogonal chromatic number of certain circulant graphs is explored. Lastly, orthogonal colourings of product graphs and Hamming graphs are studied.
For a finite group $G$ and an inverse-closed generating set $C$ of $G$, let $Aut(G;C)$ consist of those automorphisms of $G$ which leave $C$ invariant. We define an $Aut(G;C)$-invariant normal subgroup $Φ(G;C)$ of $G$ which has the property that, for any $Aut(G;C)$-invariant normal set of generators for $G$, if we remove from it all the elements of $Φ(G;C)$, then the remaining set is still an $Aut(G;C)$-invariant normal generating set for $G$. The subgroup $Φ(G;C)$ contains the Frattini subgroup $Φ(G)$ but the inclusion may be proper. The Cayley graph $Cay(G,C)$ is normal edge-transitive if $A…
A Cayley digraph $\rm{Cay}(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called a CI-digraph if for any Cayley digraph $\rm{Cay}(G,T)$ isomorphic to $\rm{Cay}(G,S)$, there is an $α\in \rm{Aut}(G)$ such that $S^α=T$. For a positive integer $m$, $G$ is said to have the $m$-DCI property if all Cayley digraphs of $G$ with out-valency $m$ are CI-digraphs. Li [The Cyclic groups with the $m$-DCI Property, European J. Combin. 18 (1997) 655-665] characterized cyclic groups with the $m$-DCI property, and in this paper, we characterize dihedral groups with the $m$-DCI property. For a dihedr…
Let $G$ be a group and let $S$ be an inverse-closed and identity-free generating set of $G$. The \emph{Cayley graph} $\Cay(G,S)$ has vertex-set $G$ and two vertices $u$ and $v$ are adjacent if and only if $uv^{-1}\in S$. Let $CAY_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs of order at most $n$. We show that $\log(CAY_d(n))\inΘ(d(\log n)^2)$, as $n\to\infty$. We also obtain some stronger results in the case $d=3$.
The strong hull property for affine irreducible Coxeter groups of rank 3
本文证明了所有秩为3的仿射不可约Coxeter群满足强壳性质，通过Coxeter复形的几何性质将凸壳分析简化为有限可管理的配置。
Groups with elements of order 8 do not have the DCI property
本文证明含有8阶元素的群不具有DCI性质，并给出具有CI但不具有DCI性质的第二个无限群族实例。
We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon--Roichman about expanding properties of random Cayley graphs. Applying the general setting we show e.g., that with high probability (in a strong exp…
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