An extended duality theorem and multiplication of several matrices

Motivated by a construction of highly expanding simple Cayley graphs of dihedral groups derived from-or induced by-highly expanding Cayley digraphs fo cyclic groups presented by F. R. K. chung, constructions of simple Cayley graphs on a semidirect product of groups and Cayley digraphs of one of its factors are suggested. In case of the non-normal factor being the cyclic group of order 2. a condition is given to derive spectral bounds of the Cayley graph of the product from those of the Cayley graph of the normal factor.     

Induced cayley graphs and d0ouble covers

Motivated by a construction of highly expanding simple Cayley graphs of dihedral groups derived from-or induced by-highly expanding Cayley digraphs fo cyclic groups presented by F. R. K. chung, constructions of simple Cayley graphs on a semidirect product of groups and Cayley digraphs of one of its factors are suggested. In case of the non-normal factor being the cyclic group of order 2. a condition is given to derive spectral bounds of the Cayley graph of the product from those of the Cayley graph of the normal factor.     

On Cayley graphs of rectangular groups

In this paper, we first give a characterization of Cayley graphs of rectangular groups. Then, vertex-transitivity of Cayley graphs of rectangular groups is considered. Further, it is shown that Cayley graphs Cay(S,C) which are automorphism-vertex-transitive, are in fact Cayley graphs of rectangular groups, if the subsemigroup generated by C is an orthodox semigroup. Finally, a characterization of vertex-transitive graphs which are Cayley graphs of finite semigroups is concluded.     

On the category of weak Cayley table morphisms between groups

Weak Cayley table functions between groups are generalized conjugacy-preserving homomorphisms, under which products of images are conjugate to images of products. There is a weak Cayley table bijection between two groups iff they have the same 2-characters. In this paper, weak Cayley table functions are augmented to include the specific conjugating elements, leading to the concept of a weak (Cayley table) morphism. If the conjugating elements are chosen subject to a crossed-product condition, then the weak morphisms between groups form a category. The forgetful functor to this category from the category of group homomorphisms is shown to possess a left adjoint. Two weak morphisms are said to be homotopic if they project to the same weak Cayley table function. As a first step in the analysis of the category of weak morphisms, the group of units of the monoid of weak morphisms homotopic to the identity automorphism of a group is described.     

On the Cayley Graphs of Brandt Semigroups

In this article, the Cayley graphs of Brandt semigroups are investigated. The basic structures and properties of this kind of Cayley graphs are given, and a necessary and sufficient condition is given for the components of Cayley graphs of Brandt semigroups to be strongly regular. As an application, the generalized Petersen graph and k-partite graph, which cannot be obtained from the Cayley graphs of groups, can be constructed as a component of the Cayley graphs of Brandt semigroups.     

Bedingungen für die Zweispiegeligkeit der Automorphismengruppen von Cayleyalgebren

In this paper we derive necessary and sufficient conditions for the bireflectionality of the automorphism group of a Cayley algebra over a field of characteristic not 2. These are of particular interest for split Cayley algebras since their automorphism groups are the Chevalley groups of type G ₂. As an application we show the bireflectionality of the automorphism groups of Cayley algebras over real closed fields.     

Discrepancy and eigenvalues of Cayley graphs

We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This affirmatively answers a question of Chung and Graham (2002) for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.     

An explicit characterization of cubic symmetric bi-Cayley graphs on nonabelian simple groups

One of the remarkable contributions in the study of symmetric Cayley graphs on nonabelian simple groups is the complete classification of such graphs that are cubic and nonnormal. This naturally motivates the study of cubic (normal and nonnormal) symmetric bi-Cayley graphs on nonabelian simple groups. In this paper, the full automorphism groups of these graphs are determined, and necessary and sufficient conditions are given for a graph being a cubic normal symmetric Cayley or bi-Cayley graph on a nonabelian simple group (one may then find many examples). As an application, we also prove that cubic symmetric Cayley graphs on nonabelian simple groups are stable.     

Finite edge-transitive Cayley graphs and rotary Cayley maps

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups.

In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

     

On 1-factorizability of Cayley graphs

In this paper we consider the existence of a 1-factorization of undirected Cayley graphs of groups of even order. We show that a 1-factorization exists for all generating sets for even order abelian groups, dihedral groups, and dicyclic groups and for all minimal generating sets for even order nilpotent groups and for D_m × Z_n. We also derive other results that are useful in considering specific Cayley graphs. These results support the conjecture that all Cayley graphs of groups of even order are 1-factorizable. If this is not the case the same result may hold for minimal generating sets.     

Automorphisms of Cayley graphs

The aim of this paper is to characterize subgroups of non-regular automorphism groups of Cayley graphs. Relations between group automorphisms, graph automorphisms and permutations of the edge set of Cayley graphs are investigated.     

The Cayley graphs of ℤ<Superscript>d</Superscript> and the limits of vertex-primitive graphs of <Emphasis Type="Italic">HA</Emphasis>-type

We study the limits of the finite graphs that admit some vertex-primitive group of automorphisms with a regular abelian normal subgroup. It was shown in [1] that these limits are Cayley graphs of the groups ?^d. In this article we prove that for each d > 1 the set of Cayley graphs of ?^d presenting the limits of finite graphs with vertex-primitive and edge-transitive groups of automorphisms is countable (in fact, we explicitly give countable subsets of these limit graphs). In addition, for d < 4 we list all Cayley graphs of ?^d that are limits of minimal vertex-primitive graphs. The proofs rely on a connection of the automorphism groups of Cayley graphs of ?^d with crystallographic groups.     

Which finitely generated Abelian groups admit isomorphic Cayley graphs?

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.     

Partial difference sets and amorphic Cayley schemes in non‐abelian 2‐groups

In this paper, we consider regular automorphism groups of graphs in the RT2 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 RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce amorphic non‐abelian Cayley schemes.     

一类新的交错群Cayley网络

本文利用交错群的Ｃａｙｌｅｙ图构作了一类互连网络,进而讨论了它的直径,容错度,容错直径和Ｈａｍｉｌｔｏｎ连通性,这些性质表明它优于利用交错群构作的网络ＡＧｎ,且相似于著名的星形网络。     

Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets

In this paper, we give a new lifting construction of “hyperbolic” type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to m-ovoids and i-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.     

On transitive Cayley graphs of strong semilattices of right (left) groups

We investigate Cayley graphs of strong semilattices of right (left) groups, of right (left) zero semigroups, and of groups. We show under which conditions they enjoy the property of automorphism vertex transitivity in analogy to Cayley graphs of groups.     

Connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups

A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two suffcient and necessary conditions for such graphs to be 1- or 2-arc-regular are given and based on the conditions, several infinite families of 1-or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.     

On Symmetric Cayley Graphs of Valency Eleven

A Cayley graph Γ=Cay(G,S)is said to be normal if G is normal in Aut Γ.In this paper,we investigate the normality problem of the connected 11-valent symmetric Cayley graphs Γ of finite nonabelian simple groups G,where the vertex stabilizer Av is soluble for A=Aut Γ and v ∈ VΓ.We prove that either Γ is normal or G=A5,A10,A54,A274,A549 or A1099.Further,11-valent symmetric nonnormal Cayley graphs of A5,A54 and A274 are constructed.This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind(of valency 11)was constructed by Fang,Ma and Wang in 2011.     

Analogues of Cayley graphs for topological groups

We define for a compactly generated totally disconnected locally compact group a graph, called a rough Cayley graph, that is a quasi-isometry invariant of the group. This graph carries information about the group structure in an analogous way to the ordinary Cayley graph for a finitely generated group. With this construction the machinery of geometric group theory can be applied to topological groups. This is illustrated by a study of groups where the rough Cayley graph has more than one end and a study of groups where the rough Cayley graph has polynomial growth. Supported by project J2245 of the Austrian Science Fund (FWF) and be an IEF Marie Curie Fellowship of the Commission of the European Union.