Cyclic Automorphism Group Research Articles

Classification of Cubic Tricirculant Nut Graphs

A nut graph is a simple graph whose adjacency matrix has the eigenvalue zero with multiplicity one such that its corresponding eigenvector has no zero entries. It is known that there exist no cubic circulant nut graphs. A bicirculant (resp. tricirculant) graph is defined as a graph that admits a cyclic group of automorphisms having two (resp. three) orbits of vertices of equal size. We show that there exist no cubic bicirculant nut graphs and we provide a full classification of cubic tricirculant nut graphs.

The $a$-numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6

The $a$-numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6

An Unexpected Cyclic Symmetry of I{mathfrak u}_n

We find and discuss an unexpected (to us) order n cyclic group of automorphisms of the Lie algebra I{mathfrak u}_n{:}{=}{mathfrak u}_n < imes {mathfrak u}_n^*, where {mathfrak u}_n is the Lie algebra of upper triangular ntimes n matrices. Our results also extend to mathfrak {gl}_{n+}^epsilon , a “solvable approximation” of mathfrak {gl}_n, as defined within.

Finite cubic graphs admitting a cyclic group of automorphism with at most three orbits on vertices

The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertex-transitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with at most three orbits on vertices either belongs to one of 5 infinite families or is isomorphic to the well-known Tutte–Coxeter graph.

Counting the number of quasiplatonic topological actions of the cyclic group on surfaces

We derive a closed formula for the number of quasiplatonic topological actions of the cyclic group on surfaces of genus two or greater. The formula implies that the number of quasiplatonic cyclic group actions is roughly one-sixth the number of regular dessins d’enfants with a cyclic group of automorphisms.

Cyclic Automorphism Groups of Graphs and Edge-Colored Graphs

In this paper we describe the automorphism groups of graphs and edge-colored graphs that are cyclic as permutation groups. In addition, we show that every such group is the automorphism group of a complete graph whose edges are colored with 3 colors, and we characterize those groups that are automorphism groups of simple graphs.

On twists of smooth plane curves

Given a smooth curve defined over a field $k$ that admits a non-singular plane model over $\overline{k}$, a fixed separable closure of $k$, it does not necessarily have a non-singular plane model defined over the field $k$. We determine under which conditions this happens and we show an example of such phenomenon. Now, even assuming that such a smooth plane model exists, we wonder about the existence of non-singular plane models over $k$ for its twists. We characterize twists possessing such models and use such characterization to improve, for the particular case of smooth plane curves, the algorithm to compute twists of non-hyperelliptic curves wrote recently down by the third author. We also show an example of a twist not admitting such non-singular plane model. As a consequence, we get explicit equations for a non-trivial Brauer-Severi surface. Finally, we obtain a theoretical result to compute all the twists of smooth plane curves with cyclic automorphism group having a $k$-model whose automorphism group is generated by a diagonal matrix. Some examples are also provided.

The hyperbolicity constant of infinite circulant graphs

Abstract If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. Deciding whether or not a graph is hyperbolic is usually very difficult; therefore, it is interesting to find classes of graphs which are hyperbolic. A graph is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we prove that infinite circulant graphs and their complements are hyperbolic. Furthermore, we obtain several sharp inequalities for the hyperbolicity constant of a large class of infinite circulant graphs and the precise value of the hyperbolicity constant of many circulant graphs. Besides, we give sharp bounds for the hyperbolicity constant of the complement of every infinite circulant graph.

Circulant Homogeneous Factorisations of Complete Digraphs $\rm\bf K_{p^d}$ with $p$ an Odd Prime

Let $\mathcal F=(\rm\bf K_{n},\mathcal P)$ be a circulant homogeneous factorisation of index $k$, that means $\mathcal P$ is a partition of the arc set of the complete digraph $\rm\bf K_n$ into $k$ circulant factor digraphs such that there exists $\sigma\in S_n$ permuting the factor circulants transitively amongst themselves. Suppose further such an element $\sigma$ normalises the cyclic regular automorphism group of these circulant factor digraphs, we say $\mathcal F$ is normal. Let $\mathcal F=(\rm\bf K_{p^d},\mathcal P)$ be a circulant homogeneous factorisation of index $k$ where $p^d$, ($d\ge 1$) is an odd prime power. It is shown in this paper that either $\mathcal F$ is normal or $\mathcal F$ is a lexicographic product of two smaller circulant homogeneous factorisations.

The classification of half-arc-regular bi-circulants of valency [formula omitted

A graph with a cyclic group of automorphisms acting semiregularly on the vertices with two orbits is called a bi-circulant. A graph is half-arc-transitive if it is both vertex-transitive and edge-transitive but not arc-transitive. Then we say that a half-arc-transitive graph is half-arc-regular if its full automorphism group acts regularly on its edges. It is known that the smallest possible valency of a half-arc-transitive bi-circulant is 6. In this paper, a classification is given of connected half-arc-regular bi-circulants of valency 6. As byproduct, we construct the first known infinite family of half-arc-transitive bi-circulants of valency 6.

A note on another construction of graphs with $4n+6$ vertices and cyclic automorphism group of order $4n$

A note on another construction of graphs with $4n+6$ vertices and cyclic automorphism group of order $4n$

On a certain family of asymmetric Riemann surfaces with the cyclic automorphism group

A compact Riemann surface X of genus g ≥ 2 is called asymmetric or pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. The order d = #(δ) of an anticonformal automorphism δ of such a surface is divisible by 4. In the particular case where d = 4, δ is a pseudo-symmetry and the surface is called pseudo-symmetric. A Riemann surface X is said to be p-hyperelliptic if it admits a conformal involution ρ for which the orbit space X/ has genus p. This notion is the particular case of so called cyclic (q,n)-gonal surface which is defined as the one admitting a conformal automorphism φ of prime order n such that X/φ has genus q. We are interested in possible values of n and q for which an asymmetric surface of given genus g ≥ 2 is (q,n)-gonal, and possible values of p for which the surface is p-hyperelliptic. Up till now, this problem was solved in the case where the surface is asymmetric and pseudo-symmetric. If an asymmetric Riemann surface X is not pseudo-symmetric then any anticonformal automorphism of X has order divisible by 2sn for s ≥ 3 and n = 1 or n being an odd prime. In this paper we give the necessary and sufficient conditions on the existence of an asymmetric Riemann surface with the full automorphism group being G = Z2sn, and we study (q,n)-gonal automorphisms and p-hyperelliptic involutions in G.

Counting surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group with motivations from string theory and QFT

Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). Recently, Koch et al. (2013) [12] gave a refined formula for counting ribbon graphs and discussed its applications to several physics problems. An important factor in this formula is the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. The aim of this paper is to give an explicit and practical formula for the number of such epimorphisms. As a consequence, we obtain an ‘equivalent’ form of Harvey's famous theorem on the cyclic groups of automorphisms of compact Riemann surfaces. Our main tool is an explicit formula for the number of solutions of restricted linear congruence recently proved by Bibak et al. using properties of Ramanujan sums and of the finite Fourier transform of arithmetic functions.

Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangulations include the Brieskorn homology spheres $\Sigma (p,q,r)$, the lens spaces $\operatorname{L} (q,1)$ and, as a limit case, $(\mathbf{S}^2 \times \mathbf{S}^1)^{\# (p-1)(q-1)}$.

Periodicity of self-injective algebras of polynomial growth

Let A be an indecomposable representation-infinite tame finite-dimensional algebra of polynomial growth over an algebraically closed field. We prove that A is a periodic algebra with respect to action of the bimodule syzygy operator if and only if A is Morita equivalent to a socle deformation of an orbit algebra Bˆ/G where Bˆ is the repetitive category of a tubular algebra B and G is an admissible infinite cyclic automorphism group of Bˆ. The main contribution in the paper is to prove the sufficiency part of this equivalence. It is known that every orbit algebra Bˆ/G of a tubular algebra B admits a presentation as an orbit algebra T(B)(r)/H of an r-fold trivial extension algebra T(B)(r) of B with respect to free action of a finite cyclic automorphism group H of T(B)(r). A significant part of the paper is devoted to explicit descriptions of the minimal projective bimodule resolutions of properly chosen ten exceptional self-injective algebras of polynomial growth and showing that all of them are periodic algebras. Then we deduce the periodicity of all algebras socle equivalent to the orbit algebras Bˆ/G=T(B)(r)/H of tubular algebras B from the periodicity of these ten exceptional algebras, using invariance of periodicity for finite Galois coverings and derived equivalences of algebras.

Mathematical Properties of the Hyperbolicity of Circulant Networks

IfXis a geodesic metric space andx1,x2,x3∈X, ageodesic triangle T={x1,x2,x3}is the union of the three geodesics[x1x2],[x2x3], and[x3x1]inX. The spaceXisδ-hyperbolic(in the Gromov sense) if any side ofTis contained in aδ-neighborhood of the union of the two other sides, for every geodesic triangleTinX. The study of the hyperbolicity constant in networks is usually a very difficult task; therefore, it is interesting to find bounds for particular classes of graphs. A network is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we obtain several sharp inequalities for the hyperbolicity constant of circulant networks; in some cases we characterize the graphs for which the equality is attained.

Strongly regular m-Cayley circulant graphs and digraphs

The first part of this paper is a survey about strongly regular graphs and digraphs admitting a semiregular cyclic group of automorphisms. In the second part, some new types of such digraphs, called uniform and almost uniform, are studied. By using partial sum families, the form of the parameters is determined and some directed strongly regular graphs derived from these partial sum families with previously unknown parameters are obtained.

Group actions of prime order on local normal rings

Let B be a Noetherian normal local ring and G Aut.B/ be a cyclic group of local automorphisms of prime order. Let A be the subring of G-invariants of B and assume that A is Noetherian. We prove that B is a monogenous A-algebra if and only if the augmentation ideal of B is principal. If in particular B is regular, we prove that A is regular if the augmentation ideal of B is principal.

Symmetry types of cyclic covers of the sphere

We examine all compact Riemann surfaces of genus greater than one which admit a cyclic group of automorphisms that yields a covering of the Riemann sphere with exactly three branch points. We determine the number of non-conjugate symmetries of each of these surfaces. For each symmetry, we determine the number of ovals it fixes and whether the orbit space under the symmetry is orientable or not. This yields the species of each symmetry and the symmetry type of each surface. Explicit defining equations of each surface and symmetry are given.

On Cayley digraphs on nonisomorphic 2‐groups

AbstractA necessary and sufficient condition is given for two Cayley digraphs X1 = Cay(G1, S1) and X2 = Cay(G2, S2) to be isomorphic, where the groups Gi are nonisomorphic abelian 2‐groups, and the digraphs Xi have a regular cyclic group of automorphisms. Our result extends that of Morris [J Graph Theory 3 (1999), 345–362] concerning p‐groups Gi, where p is an odd prime. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory