Bipartite bi-Cayley graphs over metacyclic groups of odd prime-power order

A graph $\Gamma$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $\Gamma$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $\Gamma$ be a connected bipartite bi-Cayley graph over the group $G$. In this paper, we prove that $G$ is normal in the full automorphism group ${\rm Aut}(\Gamma)$ of $\Gamma$ when $G$ is a Sylow $p$-subgroup of ${\rm Aut}(\Gamma)$. As an application, we classify half-arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $2p$. Furthermore, it is shown that there are no semisymmetric and no arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $p$.Comment: 20 pages, 1 figur

Vertex-transitive Haar graphs that are not Cayley graphs

In a recent paper (arXiv:1505.01475 ) Est\'elyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph $G(10,2)$, occurring as the graph $40$ in the Foster census of connected symmetric trivalent graphs.Comment: 9 pages, 2 figure

Stability of circulant graphs

The canonical double cover $\mathrm{D}(\Gamma)$ of a graph $\Gamma$ is the direct product of $\Gamma$ and $K_2$. If $\mathrm{Aut}(\mathrm{D}(\Gamma))=\mathrm{Aut}(\Gamma)\times\mathbb{Z}_2$ then $\Gamma$ is called stable; otherwise $\Gamma$ is called unstable. An unstable graph is nontrivially unstable if it is connected, non-bipartite and distinct vertices have different neighborhoods. In this paper we prove that every circulant graph of odd prime order is stable and there is no arc-transitive nontrivially unstable circulant graph. The latter answers a question of Wilson in 2008. We also give infinitely many counterexamples to a conjecture of Maru\v{s}i\v{c}, Scapellato and Zagaglia Salvi in 1989 by constructing a family of stable circulant graphs with compatible adjacency matrices

Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime

A regular cover of a connected graph is called {\em cyclic} or {\em dihedral} if its transformation group is cyclic or dihedral respectively, and {\em arc-transitive} (or {\em symmetric}) if the fibre-preserving automorphism subgroup acts arc-transitively on the regular cover. In this paper, we give a classification of arc-transitive cyclic and dihedral covers of a connected pentavalent symmetric graph of order twice a prime. All those covers are explicitly constructed as Cayley graphs on some groups, and their full automorphism groups are determined

Arc-transitive bicirculants

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

On 2-Fold Covers of Graphs

A regular covering projection \p\colon \tX \to X of connected graphs is $G$-admissible if $G$ lifts along \p. Denote by \tG the lifted group, and let \CT(\p) be the group of covering transformations. The projection is called $G$-split whenever the extension \CT(\p) \to \tG \to G splits. In this paper, split 2-covers are considered. Supposing that $G$ is transitive on $X$, a $G$-split cover is said to be $G$-split-transitive if all complements \bG \cong G of \CT(\p) within \tG are transitive on \tX; it is said to be $G$-split-sectional whenever for each complement \bG there exists a \bG-invariant section of \p; and it is called $G$-split-mixed otherwise. It is shown, when $G$ is an arc-transitive group, split-sectional and split-mixed 2-covers lead to canonical double covers. For cubic symmetric graphs split 2-cover are necessarily cannonical double covers when $G$ is 1- or 4-regular. In all other cases, that is, if $G$ is $s$-regular, $s=2,3$ or 5, a necessary and sufficient condition for the existence of a transitive complement \bG is given, and an infinite family of split-transitive 2-covers based on the alternating groups of the form $A_{12k+10}$ is constructed. Finally, chains of consecutive 2-covers, along which an arc-transitive group $G$ has successive lifts, are also considered. It is proved that in such a chain, at most two projections can be split. Further, it is shown that, in the context of cubic symmetric graphs, if exactly two of them are split, then one is split-transitive and the other one is either split-sectional or split-mixed.Comment: 18 pages, 3 figure

Edge-transitive bi-Cayley graphs

A graph \G admitting a group $H$ of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a {\em bi-Cayley graph\/} over $H$. Such a graph \G is called {\em normal\/} if $H$ is normal in the full automorphism group of \G, and {\em normal edge-transitive\/} if the normaliser of $H$ in the full automorphism group of \G is transitive on the edges of \G. % In this paper, we give a characterisation of normal edge-transitive bi-Cayley graphs, %which form an important subfamily of 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, `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 $6$, and answer some open questions from the literature about $2$-arc-transitive, half-arc-transitive and semisymmetric graphs

Finite edge-transitive oriented graphs of valency four: a global approach

We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of "basic" graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restirictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.Comment: 19 page

Derangement action digraphs and graphs

We study the family of \emph{derangement action digraphs}, which are a subfamily of the group action graphs introduced in [Fred Annexstein, Marc Baumslag, and Arnold L. Rosenberg, Group action graphs and parallel architectures, \emph{SIAM J. Comput.} 19 (1990), no. 3, 544--569]. For any non-empty set $X$ and a non-empty subset $S$ of \Der(X), the set of derangments of $X$, we define the derangement action digraph $\rm\overrightarrow{DA}(X;S)$ to have vertex set $X$, and an arc from $x$ to $y$ if and only if $y=x^s$ for some $s\in S$. In common with Cayley graphs and digraphs, derangement action digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on $S$ under which $\rm\overrightarrow{DA}(X;S)$ may be viewed as a simple graph of valency $|S|$, and we call such graphs derangement action graphs. Also we investigate the structural and symmetry properties of these digraphs and graphs. Several open problems are posed and many examples are given.Comment: 15 pages, 1 figur

A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two

A complete list of all connected arc-transitive asymmetric digraphs of in-valence and out-valence 2 on up to 1000 vertices is presented. As a byproduct, a complete list of all connected 4-valent graphs admitting a half-arc-transitive group of automorphisms on up to 1000 vertices is obtained. Several graph-theoretical properties of the elements of our census are calculated and discussed