Every finite group has a normal bi-Cayley graph

A graph \G with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {\em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of \Aut(\G), we say that \G is {\em normal} with respect to $H$. In this paper, we show that every finite group has a connected normal bi-Cayley graph. This improves Theorem~5 of [M. Arezoomand, B. Taeri, Normality of 2-Cayley digraphs, Discrete Math. 338 (2015) 41--47], and provides a positive answer to the Question of the above paper

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

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 $r$

Cubic vertex-transitive non-Cayley graphs of order 12p

A graph is said to be {\em vertex-transitive non-Cayley} if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive non-Cayley graphs of order $12p$, where $p$ is a prime, is given. As a result, there are $11$ sporadic and one infinite family of such graphs, of which the sporadic ones occur when $p=5$, $7$ or $17$, and the infinite family exists if and only if $p\equiv1\ (\mod 4)$, and in this family there is a unique graph for a given order.Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematic

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

Isomorphisms of Cayley graphs on nilpotent groups

Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G;S) and Cay(G';S') are connected Cayley graphs of finite valency on two nilpotent groups G and G', then every isomorphism from Cay(G;S) to Cay(G';S') factors through to a well-defined affine map from G/N to G'/N', where N and N' are the torsion subgroups of G and G', respectively. For the special case where the groups are abelian, these results were previously proved by A.A.Ryabchenko and C.Loeh, respectively.Comment: 12 pages, plus 7 pages of notes to aid the referee. One of our corollaries is already known, so a reference to the literature has been adde

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

Asymptotic Automorphism Groups of Circulant Graphs and Digraphs

We show that almost all circulant graphs have automorphism groups as small as possible. Of the circulant graphs that do not have automorphism group as small as possible, we give some families of integers such that it is not true that almost all circulant graphs whose order lies in any one of these families, are normal. That almost all Cayley (di)graphs whose automorphism group is not as small as possible are normal was conjectured by the second author, so these results provide counterexamples to this conjecture. It is then shown that there is a large family of integers for which almost every circulant digraph whose order lies in this family and that does not have automorphism group as small as possible, is normal. We additionally explore the asymptotic behavior of the automorphism groups of circulant (di)graphs that are not normal, and show that no general conclusion can be obtained.Comment: 26 page

On the scaling limit of finite vertex transitive graphs with large diameter

Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $|X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $<q$, equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If $X_n$ is only roughly transitive and $|X_n| = o\bigl({diam(X_n)^{\delta}}\bigr)$ for $\delta > 1$ sufficiently small, we prove, this time by elementary means, that $(X_n)$ converges to a circle.Comment: Final version, to appear in Combinatoric

Eigenvalues of Cayley graphs

We survey some of the known results on eigenvalues of Cayley graphs and their applications, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs