Full Automorphism Group Research Articles

Generalized Gardiner–Praeger graphs and their symmetries

A subgroup of the automorphism group of a graph acts half-arc-transitively on the graph if it acts transitively on the vertex-set and on the edge-set of the graph but not on the arc-set of the graph. If the full automorphism group of the graph acts half-arc-transitively, the graph is said to be half-arc-transitive.In 1994 Gardiner and Praeger introduced two families of tetravalent arc-transitive graphs, called the C±1 and the C±ε graphs, that play a prominent role in the characterization of the tetravalent graphs admitting an arc-transitive group of automorphisms with a normal elementary abelian subgroup such that the corresponding quotient graph is a cycle. All of the Gardiner–Praeger graphs are arc-transitive but admit a half-arc-transitive group of automorphisms. Quite recently, Potočnik and Wilson introduced the family of CPM graphs, which are generalizations of the Gardiner–Praeger graphs. Most of these graphs are arc-transitive, but some of them are half-arc-transitive. In fact, at least up to order 1000, each tetravalent half-arc-transitive loosely-attached graph of odd radius having vertex-stabilizers of order greater than 2 is isomorphic to a CPM graph.In this paper we determine the automorphism group of the CPM graphs and investigate isomorphisms between them. Moreover, we determine which of these graphs are 2-arc-transitive, which are arc-transitive but not 2-arc-transitive, and which are half-arc-transitive.

The smallest symmetric cubic graphs with given type

It is known that arc-transitive group actions on finite cubic (3-valent) graphs fall into seven classes, denoted by 1, 21, 22, 3, 41, 42 and 5, where k, k1 or k2 indicates that the action is k-arc-regular, with k2 indicating that there is no arc-reversing automorphism of order 2 (for k=2 or 4). These classes can be further subdivided into 17 sub-classes, according to the types of arc-transitive subgroups of the full automorphism group of the graph, sometimes called the ‘action type’ of the graph. In this paper, we complete the determination of the smallest graphs in each of these 17 classes (begun by Conder and Nedela in (2009) [7]).

On generalized truncations of complete graphs

For a k -regular graph Γ and a graph Υ of order k , a generalized truncation of Γ by Υ is constructed by replacing each vertex of Γ with a copy of Υ . E. Eiben, R. Jajcay and P. S parl introduced a method for constructing vertex-transitive generalized truncations. For convenience, we call a graph obtained by using Eiben et al. ’s method a special generalized truncation . In their paper, Eiben et al. proposed a problem to classify special generalized truncations of a complete graph K n by a cycle of length n − 1 . In this paper, we completely solve this problem by demonstrating that with the exception of n = 6 , every special generalized truncation of a complete graph K n by a cycle of length n − 1 is a Cayley graph of AGL(1, n ) where n is a prime power. Moreover, the full automorphism groups of all these graphs and the isomorphisms among them are determined.

On the automorphism group of doubled Grassmann graphs

In this paper, we study a class of regular graphs, which is related to the Grassmann graph. This class of graphs is called the doubled Grassman graph. The Grassmann graph is the class of graphs, which is defined similar to the Johnson graph. This time, however, we are concerned with the subspaces of a vector space, rather than with the subsets of a set. The doubled Grassmann graph is constructed from the Grassmann graph. This graph, was discovered by Biggs and Gardiner. In this paper, we determine the full automorphism group of the doubled Grassmann graph. For determining the automorphism group of the doubled Grassmann graph, we will use the methods used by Mirafzal (Proc. Indian Acad. Sci. (Math Sci.) 129(3) (2019), Art. 34, 8).

A classification of tetravalent half-arc-transitive metacirculants of 2-power orders

Metacirculants were introduced by Alspach and Parsons in 1982 and have been a rich source of various topics since then. A graph is half-arc-transitive if its full automorphism group acts transitively on vertices and edges, but not on arcs. In Feng et al. (2007), the first infinite families of tetravalent half-arc-transitive metacirculants of order 2-powers were introduced. In this paper, a complete classification is given of tetravalent half-arc-transitive metacirculants of order 2-powers.

Canonical double covers of generalized Petersen graphs, and double generalized Petersen graphs

AbstractThe canonical double cover of a graph is the direct product of and . If then is called stable; otherwise is called unstable. An unstable graph is said to be nontrivially unstable if it is connected, non‐bipartite and no two vertices have the same neighborhood. In 2008 Wilson conjectured that, if the generalized Petersen graph is nontrivially unstable, then both and are even, and either is odd and , or . In this note we prove that this conjecture is true. At the same time we determine all possible isomorphisms among the generalized Petersen graphs, the canonical double covers of the generalized Petersen graphs, and the double generalized Petersen graphs. Based on these we completely determine the full automorphism group of the canonical double cover of for any pair of integers with .

Automorphism group and fixing number of the orthogonality graph based on rank one upper triangular matrices

The orthogonality graph [Formula: see text] of a ring [Formula: see text] is the undirected graph with vertex set consisting of all nonzero two-sided zero divisors of [Formula: see text], in which for two vertices [Formula: see text] and [Formula: see text] (needless distinct), [Formula: see text] is an edge if and only if [Formula: see text]. Let [Formula: see text], [Formula: see text] be the set of all [Formula: see text] matrices over a finite field [Formula: see text], and [Formula: see text] the subset of [Formula: see text] consisting of all rank one upper triangular matrices. In this paper, we describe the full automorphism group, and using the technique of generalized equivalent canonical form of matrices, we compute the fixing number of [Formula: see text], the induced subgraph of [Formula: see text] with vertex set [Formula: see text].

Symmetric property and reliability of locally twisted cubes

The locally twisted cube LTQn is a variant of the hypercube Qn, which was introduced by Yang et al. (2005) as an interconnection network for parallel computing. The symmetry of Qn is well-known, for example, it is an edge-transitive Cayley graph. However, the symmetry of LTQn remains unclear. In this paper, we first prove that LTQn with n≥4 is isomorphic to a bi-Cayley graph of an elementary abelian 2-group Z2n−1 of order 2n−1, and then prove that the full automorphism group of LTQn with n≥4 is isomorphic to Z2n−1. These show that LTQn with n≥4 is not edge-transitive, and its full automorphism group has exactly two orbits on the vertex set of LTQn (and consequently it is not vertex-transitive and not a Cayley graph). What is more, the symmetry of LTQn with n≥4 also implies that it can be decomposed to two vertex-disjoint (n−1)-dimensional hypercubes and a perfect matching. As an application, we obtain the k-extra connectivity and (k+1)-component connectivity with k≤n−1 of LTQn, which generalize some previous works.

Vertex-minimal graphs with nonabelian $${\mathbf{2}}$$-group symmetry

A graph whose full automorphism group is isomorphic to a finite group G is called a G-graph, and we let $$\alpha (G)$$ denote the minimal number of vertices among all G-graphs. The value of $$\alpha (G)$$ has been established for numerous infinite families of groups. In this article, we expand upon the subject matter of vertex-minimal G-graphs by computing the value of $$\alpha (G)$$ when G is isomorphic to either a quasi-dihedral group or a quasi-abelian group. These results completely establish the value of $$\alpha (G)$$ when G is a member of one of the six infinite families of 2-groups that contain a cyclic subgroup of index 2.

Semisymmetric prime-valent graphs of order $$2p^3$$

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime and $$\Gamma $$ a semisymmetric prime-valent graph of order $$2p^3$$ . Then, $$\Gamma $$ is bipartite. Denote by $$\mathrm{Aut}(\Gamma )$$ the full automorphism group of $$\Gamma $$ . In Du and Wang (J Algebraic Combin 41:275–302, 2015), Wang and Du (Eur J Combin 36:393–405, 2014) and Wang et al. (Ars Math Contemp 7:40–53, 2014), the first author and Du proved that there is no prime-valent graph when $$\mathrm{Aut}(\Gamma )$$ acts unfaithfully on at least one bipart of $$\Gamma $$ . The first author in Wang (Ars Combin 133:3–15, 2017) gave a complete classification when $$\mathrm{Aut}(\Gamma )$$ acts faithfully and primitively on at least one bipart of $$\Gamma $$ , and as a result there is only one such graph, that is, cubic Gray graph. Due to these efforts, there is only one remaining case for classifying such graphs: $$\mathrm{Aut}(\Gamma )$$ acts faithfully and imprimitively on both biparts of $$\Gamma $$ , which is dealt with in this paper. We prove that there are two infinite families of such graphs. Thus, combining these results, we can get the complete classification of semisymmetric prime-valent graphs of order $$2p^3$$ .

Characterising Clifford parallelisms among Clifford-like parallelisms

We recall the notions of Clifford and Clifford-like parallelisms in a 3-dimensional projective double space. In a previous paper the authors proved that the linear part of the full automorphism group of a Clifford parallelism is the same for all Clifford-like parallelisms which can be associated to it. In this paper, instead, we study the action of such group on parallel classes thus achieving our main results on characterisation of the Clifford parallelisms among Clifford-like ones.

Edge-transitive bi-Cayley graphs

A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. This generalisation of a Cayley graph gives a class of graphs that includes many important examples such as the Petersen graph, the Gray graph and the Hoffman-Singleton graph. A bi-Cayley graph Γ over a group H is called normal if H is normal in the full automorphism group of Γ, and almost-normal edge-transitive if the normaliser of H in the full automorphism group of Γ is transitive on the edges of Γ. In this paper, we give a characterisation of almost-normal edge-transitive bi-Cayley graphs, and in particular, we give a detailed description of 2-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic p-groups. We find that under certain conditions, ‘almost-normal edge-transitive’ is the same as ‘normal’ for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most 6. Also we answer some open questions from the literature about these and related classes of graphs, posed by Li (in Proc. Amer. Math. Soc. 133 (2005)), Marušič and Potočnik (in Eur. J. Comb. 22 (2001)) and Marušič and Šparl (in J. Algebr. Comb. 28 (2008)), and we correct a major error in a recent paper by Li, Song and Wang (in J. Comb. Theory Ser. A 120 (2013)).

A classification of tetravalent non-normal Cayley graphs of order twice a prime square

A Cayley graph \(\mathrm{Cay}(G, S)\) on a group G is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of \(\mathrm{Cay}(G, S)\). In this paper, a complete classification is given of tetravalent non-normal Cayley graphs of order \(2p^2\) for each prime p.

Characterization of automorphisms of Hom-biproducts

We study certain subgroups of the full group of monoidal Hom-Hopf algebra automorphisms of a Hom-biproduct, which gives a Hom-version of Radford’s results.

Characterization of automorphisms of twisted tensor biproducts

Abstract We study certain subgroups of the full group of Hopf algebra automorphisms of twisted tensor biproducts.

Semifield Planes Admitting the Quaternion Group Q8

We discuss a well-known conjecture that the full automorphism group of a finite projective plane coordinatized by a semifield is solvable. For a semifield plane of order pN (p > 2 is a prime, 4|p − 1) admitting an autotopism subgroup H isomorphic to the quaternion group Q8, we construct a matrix representation of H and a regular set of the plane. All nonisomorphic semifield planes of orders 54 and 134 admitting Q8 in the autotopism group are pointed out. It is proved that a semifield plane of order p4, 4|p−1, does not admit SL(2, 5) in the autotopism group.

Small Directed Strongly Regular Graphs

The goal of the present paper is to provide a gallery of small directed strongly regular graphs. For each graph of order n ≤ 12 and valency k < n/2, a diagram is depicted, its relation to other small directed strongly regular graphs is revealed, the full group of automorphisms is described, and some other nice properties are given. To each graph a list of interesting subgraphs is provided as well.

Bipartite edge-transitive bi-p-metacirculants

A graph is a bi-Cayley graph over a group if the group acts semiregularly on the vertex set of the graph with two orbits. Let G be a non-abelian metacyclic p -group for an odd prime p . In this paper, we prove that if G is a Sylow p -subgroup in the full automorphism group Aut(Γ) of a graph Γ , then G is normal in Aut(Γ) . As an application, we classify the half-arc-transitive bipartite bi-Cayley graphs over G of valency less than 2 p , while the case for valency 4 was given by Zhang and Zhou in 2019. It is further shown that there are no semisymmetric or arc-transitive bipartite bi-Cayley graphs over G of valency less than p .

ON A DESIGN FROM PRIMITIVE REPRESENTATIONS OF THE FINITE SIMPLE GROUPS

In this paper we present a design construction from primitive permutation representations of a finite simple group G. The group G acts primitively onthe points and transitively on the blocks of the design. The construction has this property that with some conditions we can obtain t-design for t >=2. We examine our design for fourteen sporadic simple groups. As a result we found a 2-(176,5,4) design with full automorphism group M22.

Embeddings and Ambient Automorphisms of the Pappus Configuration

We classify embeddings (i.e., “labeled drawings”) of the Pappus configuration in projective planes over commutative fields, up to projective equivalence. Using pairs of field elements, we parameterize the space of classes of projectively equivalent embeddings, and then explicitly determine the group of ambient automorphisms (or dualities) for any given parameter pair, i.e., the subgroup of the group of all automorphisms (and dualities) of the abstract configuration that are induced by projective collineations (or dualities) leaving invariant the image under any embedding in the given class. It turns out that the existence of an ambient duality implies an ambient polarity. We show that these parameter pairs can be interpreted as pairs of cross ratios associated in a rather natural way with the embedded configuration. The number of equivalence classes of embeddings in a projective plane over a given finite field is determined. The groups that occur as full ambient groups are identified in the subgroup lattice of the full automorphism group of the abstract configuration. Finally, we use our results to understand embeddings of the Mobius–Kantor configuration.