Finite edge-transitive dihedrant graphs

In this paper, we first prove that each biquasiprimitive permutation group containing a regular dihedral subgroup is biprimitive, and then give a classification of such groups. The classification is then used to classify vertex-quasiprimitive and vertex-biquasiprimitive edge-transitive dihedrants. Moreover, a characterization of valencies of normal edge-transitive dihedrants is obtained, and some classes of examples with certain valences are constructed.     

Finite vertex-primitive and vertex-biprimitive 2-path-transitive graphs

This paper presents a classification of vertex-primitive and vertex-biprimitive 2-path-transitive graphs which are not 2-arc-transitive. The classification leads to constructions of new examples of half-arc-transitive graphs.     

Finite primitive groups and edge-transitive hypergraphs

We determine all finite primitive groups that are automorphism groups of edge-transitive hypergraphs. This gives an answer to a problem proposed by Babai and Cameron.     

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.

     

Finite edge-transitive oriented graphs of valency four with cyclic normal quotients

We study finite four-valent graphs \(\Gamma \) admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on \(\Gamma \), and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show, on the one hand, that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand, existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph \(\Gamma \) and we classify all examples. We show there are five infinite families of such pairs \((\Gamma ,G)\) and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.     

A construction of pseudo metacirculants

Metacirculants were introduced by Alspach and Parsons in 1982 and have been a rich source of various topics since then. It is known that every metacirculant is a split weak metacirculant (A graph is called (split) weak metacirculant if it has a vertex-transitive (split) metacyclic subgroup of automorphisms). We say that a split metacirculant is a pseudo metacirculant if it is not metacirculant. In this paper, an infinite family of pseudo metacirculants is constructed, and this provides a negative answer to Question A in Zhou and Zhou (2018).     

On quartic half-arc-transitive metacirculants

Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ, where ρ is (m,n)-semiregular for some integers m≥1, n≥2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ ^m has at least one fixed vertex. A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed. Both authors were supported in part by “ARRS – Agencija za znanost Republike Slovenije”, program no. P1-0285.     

On edge-transitive bi-Frobenius-metacirculants

In this paper, we classify a family of edge-transitive bi-Cayley graphs on Frobenius metacyclic groups. This provides a new construction of an infinite family of half-arc-regular bi-Cayley graphs on metacyclic groups.     

On finite self-complementary metacirculants

We prove that the automorphism group of a self-complementary metacirculant is either soluble or has \(\mathrm{A}_5\) as the only insoluble composition factor, extending a result of Li and Praeger which says the automorphism group of a self-complementary circulant is soluble. The proof involves a construction of self-complementary metacirculants which are Cayley graphs and have insoluble automorphism groups. To the best of our knowledge, these are the first examples of self-complementary graphs with this property.     

On the classification of quartic half-arc-transitive metacirculants

A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ, where ρ is (m,n)-semiregular for some integers m≥1 and n≥2, and where σ normalizes ρ, cyclically permuting the orbits of ρ in such a way that σ^m has at least one fixed vertex. In a recent paper Maruši? and the author showed that each connected quartic half-arc-transitive metacirculant belongs to one (or possibly more) of four classes of such graphs, reflecting the structure of the quotient graph relative to the semiregular automorphism ρ. One of these classes coincides with the class of the so-called tightly-attached graphs, which have already been completely classified. In this paper a complete classification of the second of these classes, that is the class of quartic half-arc-transitive metacirculants for which the quotient graph relative to the semiregular automorphism ρ is a cycle with a loop at each vertex, is given.     

Classifying a family of edge-transitive metacirculant graphs

A characterization is given of the class of edge-transitive Cayley graphs of Frobenius groups \mathbbZ_p^d:\mathbbZ_q\mathbb{Z}_{p^{d}}{:}\mathbb{Z}_{q} with p,q odd prime, of valency coprime to p. This characterization is then used to study an isomorphism problem regarding Cayley graphs, and to construct new families of half-arc-transitive graphs.     

The finite vertex-primitive and vertex-biprimitive

A complete classification is given for finite vertex-primitive and vertex-biprimitive -transitive graphs for . The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group , and an infinite family of graphs of valency 5 admitting projective symplectic groups with prime and (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.

     

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.     

Completely classifying all vertex-transitive and edge-transitive polyhedra

Recently A. Dress completed the classification of the regular polyhedra in E ³ by adding one class to the enumeration given by Grünbaum on this subject. This classification is the only systematic study of a collection of polyhedra possessing special symmetries which uses the generalized definition of a polygon allowing for skew polygons as well as planar polygons in E ³. This study gives necessary conditions for polyhedra to be vertex-transitive and edge-transitive. These conditions are restrictive enough to make the task of completely enumerating such polyhedra realizable and efficient. Examples of this process are given, and an explanation of the basic process is discussed. These new polyhedra are appearing more frequently in applications of geometry, and this examination is a beginning of the classifications of polyhedra having special symmetries even though there are many other such classes which lack this scrutiny.     

Tetravalent edge-transitive graphs of order p 2 q

A graph is called edge-transitive if its full automorphism group acts transitively on its edge set.In this paper,by using classification of finite simple groups,we classify tetravalent edge-transitive graphs of order p2q with p,q distinct odd primes.The result generalizes certain previous results.In particular,it shows that such graphs are normal Cayley graphs with only a few exceptions of small orders.