In this paper, we characterise the family of finite arc-transitive bicirculants. We show that every bicirculant is a normal r $r$ -cover an graph lies in one eight infinite families or seven sporadic graphs. Moreover, each these ‘basic’ graphs either circulant, and latter case has for some integer .

A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ, and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ. Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. Djokovič and Miller (1980) proved that there are seven types of arc-transitive group action on finite cubic graphs, characterised...

An automorphism of a graph is called quasi-semiregular if it fixes unique vertex the and its remaining cycles have same length. This kind symmetry graphs was first investigated by Kutnar, Malnič, Martínez Marušič in 2013, as generalization well-known problem regarding existence semiregular automorphisms vertex-transitive graphs. Symmetric valency three or four, admitting automorphism, been clas...

A regular edge but not vertex transitive graph is said to be semisym metric The study of semisymmetric graphs was initiated by Folkman who among others gave constructions of several in nite families such graphs In this paper a generalization of his construction for or ders a multiple of is proposed giving rise to some new families of semisymmetric graphs In particular one associated with the cy...

Let Γ be a G-symmetric graph, and let B be a nontrivial G-invariant partition of the vertex set of Γ . This paper aims to characterize (Γ ,G) under the conditions that the quotient graph ΓB is (G, 2)-arc transitive and the induced subgraph between two adjacent blocks is 2 · K2 or K2,2. The results answer two questions about the relationship between Γ and ΓB for this class of graphs. c © 2007 El...

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.

We prove that, given a finite graph ? satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of ?. Applying this result, we establish the existence infinite families graphs with certain vertex stabilizers, and classify stabilizers up to order 28 connected graphs. This sheds new light on longstanding problem classifying

Abstract If ${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue $\lambda $ of a graph X and $\alpha automorphism , then ({\mathbf v})$ also . Thus, it rather exceptional vertex-transitive to have multiplicity one. We study cubic graphs with nontrivial simple eigenvalue, discover remarkable connections arc-transitivity, regular maps, number theory.

Denote by G the set of triples (Γ, X,B), where Γ is a finite X-symmetric graph of valency val(Γ) ≥ 1, B is a nontrivial X-invariant partition of V (Γ) such that ΓB, the quotient graph of Γ with respect to B, is nonempty and Γ is not a multicover of ΓB. In this article, for any given X-symmetric graph Σ, we aim to give a sufficient and necessary condition for the existence of (Γ, X,B) ∈ G, such ...

In this paper, we show that if the number of arcs in an oriented graph −→ G (of order n) without directed cycles is sufficiently small (not greater than 2 3 n− 1), then there exist arc disjoint embeddings of three copies of −→ G into the transitive tournament TTn. It is the best possible bound.