Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s + 1)-arcs. Let p be a prime. In this article a complete classification of tetravalent s-transitive graphs of order 3p is given.

A regular covering projection is called dihedral or abelian if the covering transformation group is dihedral or abelian. A lot of work has been done with regard to the classification of arc-transitive abelian (or elementary abelian, or cyclic) covers of symmetric graphs. In this paper, we investigate arc-transitive dihedral regular covers of symmetric (arc-transitive) cubic graphs. In particula...

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s + 1)-arcs. Let p be a prime. In this article a complete classification of tetravalent s-transitive graphs of order 3p is given.

The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial 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.

Abstract: When restricted to the rank 1 and rank 2 faces, the Hasse diagram of a regular abstract 4-polytope provides a bipartite graph with a high degree of symmetry. Focusing on the case of the self-dual polytopes of type 3,q,3, I will show that the graphs obtained are 3-arc transitive cubic graphs. Also, given any 3-arc transitive cubic graph, I will discuss when it is possible to consider t...

A graph is vertex-transitive if its automorphism group acts transitively on its vertices. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the cubic vertextransitive non-Cayley graphs of order 8p are classified for each prime p. It follows from this classification that there are two sporadic and two infini...

A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.

Let $G$ be a permutation group on set $\Omega$ and recall that base for is subset of such its pointwise stabiliser trivial. In recent paper, Burness Giudici introduced the Saxl graph $G$, denoted $\Sigma(G)$, with vertex two vertices adjacent if they form base. If transitive, then $\Sigma(G)$ vertex-transitive it natural to consider valency (which we refer as $G$). this paper present general me...

An undirected graph is a treelike comparability graph if it admits a transitive orientation such that its transitive reduction is a tree. We show that treelike comparability graphs are distance hereditary. Utilizing this property, we give a linear time recognition algorithm. We then characterize permutation graphs that are treelike. Finally, we consider the Partitioning into Bounded Cliques pro...