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

Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs

A finite simple graph is called a bi-Cayley graph over a group $H$ if it has a semiregular automorphism group, isomorphic to $H,$ which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been classified recently by Feng and Zhou (Europ. J. Combin. 36 (2014), 679--693). In this paper we consider the latter class of graphs and select those in the class which are also arc-transitive. Furthermore, such a graph is called $0$-type when it is bipartite, and the bipartition classes are equal to the two orbits of the respective semiregular automorphism group. A $0$-type graph can be represented as the graph $\mathrm{BCay}(H,S),$ where $S$ is a subset of $H,$ the vertex set of which consists of two copies of $H,$ say $H_0$ and $H_1,$ and the edge set is $\{\{h_0,g_1\} : h,g \in H, g h^{-1} \in S\}$. A bi-Cayley graph $\mathrm{BCay}(H,S)$ is called a BCI-graph if for any bi-Cayley graph $\mathrm{BCay}(H,T),$ $\mathrm{BCay}(H,S) \cong \mathrm{BCay}(H,T)$ implies that $T = h S^\alpha$ for some $h \in H$ and $\alpha \in \mathrm{Aut}(H)$. It is also shown that every cubic connected arc-transitive $0$-type bi-Cayley graph over an abelian group is a BCI-graph

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

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

Which Haar graphs are Cayley graphs?

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\mathrm{Aut }G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$.Comment: 13 pages, 2 figure

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 $r$

A classification of nilpotent 3-BCI groups

Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph \bcay(G,S) is the graph whose vertex set is $G \times \{0,1\}$ and edge set is $\{\{(x,0),(s x,1)\} : x \in G, s\in S \}$. A bi-Cayley graph \bcay(G,S) is called a BCI-graph if for any bi-Cayley graph \bcay(G,T), \bcay(G,S) \cong \bcay(G,T) implies that $T = g S^\alpha$ for some $g \in G$ and \alpha \in \aut(G). A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in the form $U \times V,$ where $U$ is a homocyclic group of odd order, and $V$ is trivial or one of the groups $\Z_{2^r},$ $\Z_2^r$ and \Q_8

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

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

On groups all of whose Haar graphs are Cayley graphs

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple bipartite graph $\Sigma$ such that ${\rm Aut}(\Sigma)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(\Sigma)$ and the $H$-orbits are equal to the bipartite sets of $\Sigma$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs (a group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian). As an application, it is also shown that every non-solvable group has a Haar graph which is not a Cayley graph.Comment: 17 page