F. Aguiló, M. A. Fiol, S. Pérez
Abelian Cayley digraphs can be constructed by using a generalization to $Z^n$ of the concept of congruence in $Z$. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have asymptotically large number of vertices as the diameter increases. Up to now, the best known asymptotically dense results were all non-constructive.
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Divisible design digraphs which can be obtained as Cayley digraphs are studied. A characterization of divisible design Cayley digraphs in terms of the generating sets is given. Further, we give several constructions of divisible design Cayley digraphs and classify divisible design Cayley digraphs on $v \le 27$ vertices.
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。
Higher diameters of Cayley digraphs
本文定义了Cayley有向图的更高直径概念，并研究了其性质。
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。
Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.
Researchers in the past have studied eigenvalues of Cayley digraphs or graphs. We are interested in characterizing Cayley digraphs on a finite Abelian group G whose eigenvalues are algebraic integers in a given number field K. And we succeed in finding a method to do so by proving Theorem 1. Also, the number of such Cayley digraphs is computed.
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.
Let $C_{d,k}$ be the largest number of vertices in a Cayley digraph of degree $d$ and diameter $k$, and let $BC_{d,k}$ be the largest order of a bipartite Cayley digraph for given $d$ and $k$. For every degree $d\geq2$ and for every odd $k$ we construct Cayley digraphs of order $2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ and diameter at most $k$, where $k\ge 3$, and bipartite Cayley digraphs of order $2(k-1)\left(\lfloor\frac{d}{2}\rfloor\right)^{k-1}$ and diameter at most $k$, where $k\ge 5$. These constructions yield the bounds $C_{d,k} \ge 2k\left(\lfloor\frac{d}{2}\rfloor\right)^k$ for od…
A digraph X=(V, E) is max-λ, if λ(X)=δ(X). A digraph X is super-λif every minimum cut of X is either the set of inarcs of some vertex or the set of outarcs of some vertex. In this paper, we will prove that for all but a few exceptions, the strongly connected mixed Cayley digraphs are max-λand super-λ.
We comment on the paper "Extremal Cayley digraphs of finite Abelian groups" [Intercon. Networks 12 (2011), no. 1-2, 125--135]. In particular, we give some counterexamples to the results presented there, and provide a correct result for degree two.
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。
Suppose G is a nilpotent, finite group. We show that if {a,b} is any 2-element generating set of G, then the corresponding Cayley digraph Cay(G;a,b) has a hamiltonian path. This implies there is a hamiltonian path in every connected Cayley graph on G that has valence at most 4.
Let X=(V, E) be a digraph. X is maximally connected, if κ(X)=δ(X). X is maximally arc-connected, if λ(X)=δ(X). And X is super arc-connected, if every minimum arc-cut of X is either the set of inarcs of some vertex or the set of outarcs of some vertex. In this paper, we will prove that the strongly connected Bi-Cayley digraphs are maximally connected and maximally arc-connected, and the most of strongly connected Bi-Cayley digraphs are super arc-connected.
We construct an infinite family of connected, 2-generated Cayley digraphs Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the generators a and b are arbitrarily large. We also prove that if G is any finite group with |[G,G]| < 4, then every connected Cayley digraph on G has a hamiltonian path (but the conclusion does not always hold when |[G,G]| = 4 or 5).
We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\to\infty$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n \log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the probability converges exponentially fast to 1.
We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley graphs and digraphs of diameter $k$ and degree $d$. Another method yields, for sufficiently large $k$ and infinitely many values of $d$, Cayley graphs and digraphs of diameter $k$ and degree $d$ whose order is exponentially larger in $k$ than any previously constructed. In the directed case, these are within a linear factor in $k$ of the Moore bound.
Let $Γ=\mathrm{Cay}(G,S)$ be a Cayley digraph on a group $G$ and let $A=\mathrm{Aut}(Γ)$. The Cayley index of $Γ$ is $|A:G|$. It has previously been shown that, if $p$ is a prime, $G$ is a cyclic $p$-group and $A$ contains a noncyclic regular subgroup, then the Cayley index of $Γ$ is superexponential in $p$. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if $p$ is an odd prime and $G$ is abelian but not cyclic, and has order a power of $p$ at least $p^3$, then there is a Cayley digraph $Γ$ on $G$ whose …
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。
The symmetric group is generated by σ = (1 2 ... n) and τ = (1 2). We answer an open problem of Nijenhuis and Wilf by constructing a Hamilton path in the directed Cayley graph for all n, and a Hamilton cycle for odd n.
A finite group $G$ is a "non-DCI group" if there exist subsets $S_1$ and $S_2$ of $G$, such that the associated Cayley digraphs $C\overrightarrow{ay}(G;S_1)$ and $C\overrightarrow{ay}(G;S_2)$ are isomorphic, but no automorphism of $G$ carries $S_1$ to $S_2$. Furthermore, $G$ is a "non-CI group" if the subsets $S_1$ and $S_2$ can be chosen to be closed under inverses, so we have undirected Cayley graphs $Cay(G;S_1)$ and $Cay(G;S_2)$. We show that if $p$ is a prime number, and the elementary abelian $p$-group $(\mathbb{Z}_p)^r$ is a non-DCI group, then $(\mathbb{Z}_p)^{r+3}$ is a non-CI group. I…
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。
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…
Hamilton cycles in generalized dihedral Cayley graphs and digraphs
本文证明了广义二面体群上连通Cayley有向图存在有向Hamilton圈，证实了1978年的猜想，关键新工具是三次Haar图的三重和集覆盖定理。
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。
Cayley hyper-digraphs and Cayley hypermaps
本文研究Cayley超有向图并构造高对称性的Cayley超地图，提出一般理论以促进对Cayley超地图的系统研究。
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。
Let $Γ=$Cay$(G,T)$ be a Cayley digraph over a finite Abelian group $G$ with respect the generating set $T\not\ni0$. $Γ$ has order ord$(Γ)=|G|=n$ and degree deg$(Γ)=|T|=d$. Let $k(Γ)$ be the diameter of $Γ$ and denote $κ(d,n)=\min\{k(Γ):~\textrm{ord}(Γ)=n,\textrm{deg}(Γ)=d\}$. We give a closed expression, $\ell(d,n)$, of a tight lower bound of $κ(d,n)$ by using the so called {\em solid density} introduced by Fiduccia, Forcade and Zito. A digraph $Γ$ of degree $d$ is called {\em tight} when $k(Γ)=κ(d,|Γ|)=\ell(d,|Γ|)$ holds. Recently, the {\em Dilating Method} has been developed to derive a sequ…
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,…
In this paper we show that almost all Cayley digraphs have automorphism group as small as possible; that is, they are digraphical regular representations (DRRs). More precisely, we show that as $r$ tends to infinity, for every finite group $R$ of order $r$, out of all possible Cayley digraphs on $R$ the proportion whose automorphism group is as small as possible tends to $1$. This proves a natural conjecture first proposed in $1982$ by Babai and Godsil.
A digraph D is the pattern of a matrix M when D has an arc ij if and only if the ij-th entry of M is nonzero. Study the relationship between unitary matrices and their patterns is motivated by works in quantum chaology and quantum computation. In this note, we prove that if a Cayley digraph is a line digraph then it is the pattern of a unitary matrix. We prove that for any finite group with two generators there exists a set of generators such that the Cayley digraph with respect to such a set is a line digraph and hence the pattern of a unitary matrix.
Counting fixed-point-free Cayley permutations
通过双种类方法得到Cayley排列的函数有向图的微分方程，推导出无不动点Cayley排列的显式公式，并证明其比例趋于1/e，与排列和自映射一致。
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。
Detecting Graphical and Digraphical Regular Representations in groups of squarefree order
本文精确刻画了无平方因子阶群中，哪些群满足DRR-检测或GRR-检测性质，即通过自同构固定子集的条件充分性来判定正则表示的存在性。
Normal and non-normal Cayley digraphs on cyclic and dihedral groups
本文研究循环群与二面体群上兼具正规性及非正规正则同构子群的 Cayley 有向图，证明循环群均不构成 NNND/NNN-群，并给出二面体群的完整阶数判定。
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…
Token graphs of Cayley graphs as lifts
本文提出将凯莱图的k-令牌图表示为电压图提升的一般方法，据此构造循环图和Johnson图的线图，并推导其谱，同时可推广到拉普拉斯矩阵及凯莱有向图的令牌有向图。
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…
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.
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 …
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…
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 explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-Sárközy theorem on squares in sets of integers with positive density, and the study of triangles (also called $2$-simplices) in finite fields. Among other results we show that if $\mathbb{F}_q$ is the finite field of odd order $q$, then every matrix in $Mat_d(\mathbb{F}_q), d \geq 2$ is the sum of a certain (finite) number of orthogonal matrices, this number depending only on $d$, the size of the matrix…
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…
An integer distance digraph is the Cayley graph $Γ(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to some finite subset $S \subseteq \mathbb{Z}$. The domination ratio of $Γ(\mathbb{Z},S)$ is the minimum density of a dominating set in $Γ(\mathbb{Z},S)$. We establish some basic results on the domination ratio of $Γ(\mathbb{Z},S)$ and precisely determine it when $S=\{s,t\}$ with $s$ dividing $t$.
The WL-rank of a digraph $Γ$ is defined to be the rank of the coherent configuration of $Γ$. We construct a new infinite family of strictly Deza Cayley graphs for which the WL-rank is equal to the number of vertices. The graphs from this family are divisible design and integral.
We obtain isoperimetric stability theorems for general Cayley digraphs on $\mathbb{Z}^d$. For any fixed $B$ that generates $\mathbb{Z}^d$ over $\mathbb{Z}$, we characterise the approximate structure of large sets $A$ that are approximately isoperimetric in the Cayley digraph of $B$: we show that $A$ must be close to a set of the form $kZ \cap \mathbb{Z}^d$, where for the vertex boundary $Z$ is the conical hull of $B$, and for the edge boundary $Z$ is the zonotope generated by $B$.
Lin--Lu--Yau Ricci Curvature of Digraphs via Optimal Transport Couplings
本文通过显式最优耦合构造，推导了强连通局部有限有向图的 Lin--Lu--Yau Ricci 曲率的可计算公式，统一了若干已知结果，并刻画了零曲率弧，最后应用于计算几类有向 Cayley 图的曲率。
Two Arc-Disjoint Hamiltonian Paths in Finite Two-Generated Abelian Cayley Digraphs
证明了有限阿贝尔群上两个不同非零生成元的Cayley有向图存在两条弧不交哈密顿路径，通过切割反射定理和扇区填充不等式解决。
Arc-disjoint Hamiltonian paths in Cayley digraphs with a normal difference subgroup
本文研究了具有正规差子群的Cayley有向图中弧不相交哈密顿路径的存在性，给出了精确判据，并证明所有此类图（包括有限阿贝尔群的双生成元Cayley图）均存在两条这样的路径，方法基于哈密顿割谱的距离公式和几何限制定理。
Generalized quaternion groups with the m-DCI property
本文研究广义四元数群Q_{4n}的m-DCI性质，证明了当1≤m≤2n-1时n必为奇数且不被任何不超过m-1的素数的平方整除，并给出n为素数幂时的完整刻画。
Enumeration of dicirculant digraphs
本文利用 Pólya 计数定理，对阶数 $4p$（$p$ 为素数）的双循环有向图及连通情形进行同构类计数，并给出各出度 $k$ 下的数量。
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.
Cycle tilings and $H$-factors in directed graphs
该研究提出了有向图中循环覆盖和H-因子的最小度数条件，并证明了关于传递竞赛因子的渐近精确Ore型结果。
Constructing linear codes from digraphs and groups
本文对凯莱码构造提出两种推广（图码与有向图码），分析所用（有向）图扩展性质与码参数的关系，改进了相关结果，构造出无限好有向图码族并提出开放问题。
We use the \emph{unit-graphs} and the \emph{special unit-digraphs} on matrix rings to show that every $n \times n$ nonzero matrix over $\Bbb F_q$ can be written as a sum of two $\operatorname{SL}_n$-matrices when $n>1$. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and prove that if $X$ is a subset of $\operatorname{Mat}_2 (\Bbb F_q)$ with size $|X| > \frac{2 q^3 \sqrt{q}}{q - 1}$, then $X$ contains at least two distinct matrices whose difference has determinant $α$ for any $α\in \Bbb F_q^{\ast}$. Using this result we also prove a …
Algebraic Constructions for the Digraph Routing Problems
该研究利用拟群、置换群等代数工具研究正则有向图路由的最优性质，给出重要图的紧凑代数定义，并发现广群可由不相交置换唯一表示。
The largest order $n(d,k)$ of a graph of maximum degree $d$ and diameter $k$ cannot exceed the Moore bound, which has the form $M(d,k)=d^k - O(d^{k-1})$ for $d\to\infty$ and any fixed $k$. Known results in finite geometries on generalised $(k+1)$-gons imply, for $k=2,3,5$, the existence of an infinite sequence of values of $d$ such that $n(d,k)=d^k - o(d^k)$. This shows that for $k=2,3,5$ the Moore bound can be asymptotically approached in the sense that $n(d,k)/M(d,k)\to 1$ as $d\to\infty$; moreover, no such result is known for any other value of $k\ge 2$. The corresponding graphs are, howeve…
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 $p\neq 2$.
We investigate Cayley graphs of finite semigroups and monoids. First, we look at semigroup digraphs, i.e., directed Cayley graphs of semigroups, and give a Sabidussi-type characterization in the case of monoids. We then correct a proof of Zelinka from '81 that characterizes semigroup digraphs with outdegree $1$. Further, answering a question of Knauer and Knauer, we construct for every $k\geq 2$ connected $k$-outregular non-semigroup digraphs. On the other hand, we show that every sink-free directed graph is a union of connected components of a monoid digraph. Second, we consider monoid graphs…
We demonstrate that certain Johnson-type bounds are asymptotically exact for a variety of classes of codes, namely, constant-composition codes, nonbinary constant-weight codes and multiply constant-weight codes. This was achieved via an interesting application of the theory of decomposition of edge-colored digraphs.
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