Mike Develin
Following work of Rieffel, we define the Cayley compactification of an abelian group with specified generating set. We investigate its structure using methods from discrete geometry and commutative algebra.
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It is shown that distance powers of an integral Cayley graph over an abelian group are again integral Cayley graphs over that group. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.
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…
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 graph is said to be a bi-Cayley graph over a group H if it admits H as a group of automorphisms acting semiregularly on its vertices with two orbits. A non-abelian group is called an inner-abelian group if all of its proper subgroups are abelian. In this paper, we complete the classification of connected cubic edge-transitive bi-Cayley graphs over inner-abelian p-groups for an odd prime p.
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.
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.
Let $A$ be an abelian group and let $ι$ be the automorphism of $A$ defined by $i:a\mapsto a^{-1}$. A Cayley graph $Γ=\mathrm{Cay}(A,S)$ is said to have an automorphism group \emph{as small as possible} if $\mathrm{Aut}(Γ)= A\rtimes\langle i\rangle$. In this paper, we show that almost all Cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of Babai and Godsil.
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…
Let $G$ denote a finite abelian group with identity 1 and let $S$ denote an inverse-closed subset of $G \setminus {1}$, which generates $G$ and for which there exists $s \in S$, such that $\la S \setminus \{s,s^{-1}\} \ra \ne G$. In this paper we obtain the complete classification of distance-regular Cayley graphs $\cay(G;S)$ for such pairs of $G$ and $S$.
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…
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 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.
The present work is devoted to characterize the family of symmetric undirected Cayley graphs over finite Abelian groups for degrees 4 and 6.
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。
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.
Strongly regular Cayley graphs with Paley parameters over abelian groups of rank 2 were studied in [J.A Davis, Partial difference sets in p-groups, Arch.Math.63 (1994) 103-110; K.H Leung, S.L. Ma, Partial difference sets with Paley parameters, Bull. London Math Soc. 27 (1995) 553-564]. It was shown that such graphs exist iff the corresponding group is isomorphic to ${\mathbb Z}_{p^n} \oplus {\mathbb Z}_{p^n}$, where $p$ is an odd prime. In this paper we classify all strongly regular Cayley graphs over this group using Schur rings method. As a consequence we obtain a complete classification of …
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.
We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.
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.
It is shown that a Cayley multigraph over a group $G$ with generating multiset $S$ is integral (i.e., all of its eigenvalues are integers) if $S$ lies in the integral cone over the boolean algebra generated by the normal subgroups of $G$. The converse holds in the case when $G$ is abelian. This in particular gives an alternative, character theoretic proof of a theorem of Bridges and Mena (1982). We extend this result to provide a necessary and sufficient condition for a Cayley multigraph over a Hamiltonian group to be integral, in terms of character sums and the structure of the generating set…
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…
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.
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。
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。
Using methods of Marklof and Strömbergsson we establish several limit laws for metric parameters of random Cayley graphs of finite abelian groups with respect to a randomly chosen set of generators of a fixed size. Doing so we settle a conjecture of Amir and Gurel-Gurevich.
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…
In this paper, generalized Cayley graphs are studied. It is proved that every generalized Cayley graph of order 2p is a Cayley graph, where p is a prime. Special attention is given to generalized Cayley graphs on Abelian groups. It is proved that every generalized Cayley graph on an Abelian group with respect to an automorphism which acts as inversion is a Cayley graph if and only if the group is elementary Abelian 2-group, or its Sylow 2-subgroup is cyclic. Necessary and sufficient conditions for a generalized Cayley graph to be unworthy are given.
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。
Perfect codes in Cayley graphs of abelian groups
本文研究了阿贝尔群的柯西图中的完美码和全完美码，证明了若干相关结果。
We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new families of mixed graphs. For every fixed value of the degree, these families have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are show…
Quotients of polynomial rings and regular t-balanced Cayley maps on abelian groups
本文通过建立多项式环的商与阿贝尔群上正则 t-平衡凯莱地图之间的联系，提出新的分类方法，并给出阿贝尔 2-群上 RBCM_t 的完整分类。
A finite simple graph is called a bi-Cayley graph over a group $H$ if it has a semiregular automorphism group, isomorphic to $H,$ which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014), 679--693). In this paper we consider the latter class of graphs and select those in the class which are also arc-transitive. Furthermore, such a graph is called $0$-type when it is bipartite, and the bipartition classes are equal to the two orbits of the respective semiregular automorphism g…
Perfect state transfer (PST) has great significance due to its applications in quantum information processing and quantum computation. In this paper we present a characterization on connected simple Cayley graph $Γ={\rm Cay}(G,S)$ having PST. We show that many previous results on periodicity and existence of PST of circulant graphs (where the underlying group $G$ is cyclic) and cubelike graphs ($G=(\mathbb{F}_2^n,+)$) can be derived or generalized to arbitrary abelian case in unified and more simple ways from our characterization. We also get several new results including answers on some probl…
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