Semiregular Automorphism Research Articles

Cubic Vertex-Transitive Graphs Admitting Automorphisms of Large Order

A connected graph of order n admitting a semiregular automorphism of order n/k is called a k-multicirculant. Highly symmetric multicirculants of small valency have been extensively studied, and several classification results exist for cubic vertex- and arc-transitive multicirculants. In this paper, we study the broader class of cubic vertex-transitive graphs of order n admitting an automorphism of order n/3 or larger that may not be semiregular. In particular, we show that any such graph is either a k-multicirculant for some k le 3, or it belongs to an infinite family of graphs of girth 6.

Prime-valent symmetric graphs with a quasi-semiregular automorphism

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malnič, Martínez and Marušič in 2013, as a generalization of the well-known problem regarding existence of semiregular automorphisms in vertex-transitive graphs. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers (Feng et al., 2019 [11]) and (Yin and Feng, 2021 [42]).Let Γ be a connected symmetric graph of prime valency p≥5 admitting a quasi-semiregular automorphism. In this paper, it is proved that either Γ is a connected Cayley graph Cay(M,S) such that M is a 2-group admitting a fixed-point-free automorphism of order p with S as an orbit of involutions, or Γ is a normal N-cover of a T-arc-transitive graph of valency p admitting a quasi-semiregular automorphism, where T is a non-abelian simple group and N is a nilpotent group. Further, for p=5 a complete classification of graphs Γ such that either Aut(Γ) has a solvable arc-transitive subgroup or Γ is T-arc-transitive with T a non-abelian simple group is given. Finally, a construction of an infinite family of symmetric graphs admitting a quasi-semiregular automorphism and having nonsolvable automorphism group is given.

Normality of one-matching semi-Cayley graphs over finite abelian groups with maximum degree 3

A graph $\Gamma$ 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 $\Gamma$ is normal if $G$ is a normal subgroup of ${\rm Aut}(\Gamma)$. 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 nonnormal graphs.

Arc-transitive graphs of valency twice a prime admit a semiregular automorphism

We prove that every finite arc-transitive graph of valency twice a prime admits a nontrivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant Conjecture, which states that all finite vertex-transitive digraphs admit such automorphisms.

A classification of cubic connected bi-dicirculants

A graph is said to be a bi-Cayley graph over a group H if it admits H as a semiregular automorphism group with two vertex-orbits. A bi-dicirculant is a bi-Cayley graph over a dicyclic group. In this paper, a classification of connected cubic bi-dicirculants is given. As byproducts, we show that every connected cubic vertex-transitive bi-dicirculant is Cayley, and up to isomorphism, there are two connected cubic edge-transitive bi-dicirculants, of which one has order 16, and the other has order 24.

On the Laplacian and signless Laplacian polynomials of graphs with semiregular automorphisms

A graph $$\varGamma $$ is called n-Cayley graph over a group G if $$\mathrm{Aut}(\varGamma )$$ has a semiregular subgroup isomorphic to G with n orbits (of equal size). In this paper, we give a decomposition of the Laplacian and signless Laplacian polynomials of n-Cayley graphs in terms of irreducible representations of G. Also, we construct several families of graphs with integral Laplacian and signless Laplacian spectrum.

On Certain Edge-Transitive Bicirculants

A graph $\Gamma$ of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding semiregular automorphism is a cycle, have been studied, at least for the few smallest possible valences. For valences $3$, $4$ and $5$, where the corresponding bicirculants are called generalized Petersen graphs, Rose window graphs and Tabač jn graphs, respectively, all edge-transitive members have been classified. While there are only 7 edge-transitive generalized Petersen graphs and only 3 edge-transitive Tabač jn graphs, infinite families of edge-transitive Rose window graphs exist. The main theme of this paper is the question of the existence of such bicirculants for higher valences. It is proved that infinite families of edge-transitive examples of valence $6$ exist and among them infinitely many arc-transitive as well as infinitely many half-arc-transitive members are identified. Moreover, the classification of the ones of valence $6$ and girth $3$ is given. As a corollary, an infinite family of half-arc-transitive graphs of valence $6$ with universal reachability relation, which were thus far not known to exist, is obtained.

Semiregular Automorphisms in Vertex-Transitive Graphs of Order $3p^2$

It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally $2$-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. It is the purpose of this paper to prove that vertex-transitive graphs of order $3p^2$, where $p$ is a prime, contain semiregular automorphisms.

Semiregular automorphisms in vertex-transitive graphs with a solvable group of automorphisms

It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally 2-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. The known affirmative answers for graphs with primitive and quasiprimitive groups of automorphisms suggest that solvable groups need to be considered if one is to hope for a complete solution of this conjecture. It is the purpose of this paper to present an overview of known results and suggest possible further lines of research towards a complete solution of the problem.

Normal edge-transitive and -arc-transitive semi-Cayley graphs

ABSTRACTA graph 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. In this paper, definition of normal edge-transitive, arc-transitive and -arc-transitive semi-Cayley graphs are given. The main properties of such semi-Cayley graphs are shown and a condition for a graph to be a normal edge-transitive, arc-transitive, or -arc-transitive semi-Cayley will be presented.

Trivalent vertex-transitive bi-dihedrants

A graph is said to be a bi-Cayley graph over a group H if it admits H as a semiregular automorphism group with two vertex-orbits. A bi-dihedrant is a bi-Cayley graph over a dihedral group. In this paper, it is shown that every connected trivalent edge-transitive bi-dihedrant is also vertex-transitive, and then we present a classification of trivalent arc-transitive bi-dihedrants, and study the Cayley property of trivalent vertex-transitive bi-dihedrants.

Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs

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 group. A $0$-type graph can be represented as the graph $\bc(H,S),$ where $S$ is a subset of $H,$ the vertex set of which consists of two copies of $H,$ say $H_0$ and $H_1,$ and the edge set is $\{ \{h_0,g_1\} : h,g \in H, g h^{-1} \in S\}$. A bi-Cayley graph $\bc(H,S)$ is called a BCI-graph if for any bi-Cayley graph $\bc(H,T),$ $\bc(H,S) \cong \bc(H,T)$ implies that $T = h S^\alpha$ for some $h \in H$ and $\alpha \in \aut(H)$. It is also shown that every cubic connected arc-transitive $0$-type bi-Cayley graph over an abelian group is a BCI-graph.

Semiregular Automorphisms of Cubic Vertex-Transitive Graphs and the Abelian Normal Quotient Method

We characterise connected cubic graphs admitting a vertex-transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.

Vertex-transitive generalized Cayley graphs which are not Cayley graphs

The concept of generalized Cayley graphs was introduced by Marušič et al. (1992), where it was asked if there exists a vertex-transitive generalized Cayley graph which is not a Cayley graph. In this paper the question is answered in the affirmative with a construction of two infinite families of such graphs. It is also proven that every generalized Cayley graph admits a semiregular automorphism.

Classification of Symmetric Tabačjn Graphs

A bicirculant is a graph admitting an automorphism whose cyclic decomposition consists of two cycles of equal length. In this paper we introduce the Tabaă?jn graphs, a family of pentavalent bicirculants which are a natural generalization of generalized Petersen graphs obtained from them by adding two additional perfect matchings between the two orbits of a semiregular automorphism. The main result is the classification of symmetric Tabaă?jn graphs. In particular, it is shown that the only such graphs are the complete graph $$K_{6}$$K6, the complete bipartite graph minus a perfect matching $$K_{6,6}-6K_2$$K6,6-6K2 and the icosahedron graph.

Arc-transitive graphs of valency 8 have a semiregular automorphism

One version of the polycirculant conjecture states that every vertex-transitive graph has a non-identity semiregular automorphism that is, a non-identity automorphism whose cycles all have the same length. We give a proof of the conjecture in the arc-transitive case for graphs of valency 8, which was the smallest open valency.

Semiregular automorphisms of edge-transitive graphs

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism: a non-trivial automorphism whose cycles all have the same length. In this paper, we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.

A construction for modified generalized Hadamard matrices using QGH matrices

Let G be a group of order mu and U a normal subgroup of G of order u. Let G/U = {U 1,U 2, . . . ,U m } be the set of cosets of U in G. We say a matrix H = [h ij ] of order k with entries from G is a quasi-generalized Hadamard matrix with respect to the cosets G/U if $${\sum_{1\le t \le k} h_{it}h_{jt}^{-1} = \lambda_{ij1}U_1+\cdots+\lambda_{ijm}U_m (\exists\lambda_{ij1},\ldots, \exists \lambda_{ijm} \in \mathbb{Z})}$$ for any i ? j. On the other hand, in our previous article we defined a modified generalized Hadamard matrix GH(s, u, ?) over a group G, from which a TD ? (u?, u) admitting G as a semiregular automorphism group is obtained. In this article, we present a method for combining quasi-generalized Hadamard matrices and semiregular relative difference sets to produce modified generalized Hadamard matrices.

Vertex-transitive expansions of (1, 3)-trees

A nonidentity automorphism of a graph is said to be semiregular if all of its orbits are of the same length. Given a graph X with a semiregular automorphism γ , the quotient of X relative to γ is the multigraph X / γ whose vertices are the orbits of γ and two vertices are adjacent by an edge with multiplicity r if every vertex of one orbit is adjacent to r vertices of the other orbit. We say that X is an expansion of X / γ . In [J.D. Horton, I.Z. Bouwer, Symmetric Y -graphs and H -graphs, J. Combin. Theory Ser. B 53 (1991) 114–129], Horton and Bouwer considered a restricted sort of expansions (which we will call ‘strong’ in this paper) where every leaf of X / γ expands to a single cycle in X . They determined all cubic arc-transitive strong expansions of simple (1, 3)-trees, that is, trees with all of their vertices having valency 1 or 3, thus extending the classical result of Frucht, Graver and Watkins (see [R. Frucht, J.E. Graver, M.E. Watkins, The groups of the generalized Petersen graphs, Proc. Cambridge Philos. Soc. 70 (1971) 211–218]) about arc-transitive strong expansions of K 2 (also known as the generalized Petersen graphs). In this paper another step is taken further by considering the possible structure of cubic vertex-transitive expansions of general (1,3)-multitrees (where vertices with double edges are also allowed); thus the restriction on every leaf to be expanded to a single cycle is dropped.

On ${\rm STD}_6[18,3]$'s and ${\rm STD}_7[21,3]$'s Admitting a Semiregular Automorphism Group of Order 9

We characterize symmetric transversal designs ${\rm STD}_{\lambda}[k,u]$'s which have a semiregular automorphism group $G$ on both points and blocks containing an elation group of order $u$ using the group ring ${\bf Z}[G]$. Let $n_\lambda$ be the number of nonisomorphic ${\rm STD}_{\lambda}[3\lambda,3]$'s. It is known that $n_1=1,\ n_2=1,\ n_3=4, n_4=1$, and $n_5=0$. We classify ${\rm STD}_6[18,3]$'s and ${\rm STD}_7[21,3]$'s which have a semiregular noncyclic automorphism group of order 9 on both points and blocks containing an elation of order 3 using this characterization. The former case yields exactly twenty nonisomorphic ${\rm STD}_6[18,3]$'s and the latter case yields exactly three nonisomorphic ${\rm STD}_7[21,3]$'s. These yield $n_6\geq20$ and $n_7\geq 5$, because B. Brock and A. Murray constructed two other ${\rm STD}_7[21,3]$'s in 1991. We used a computer for our research.