Related papers: Arc transitive circulants

A graph is edge-transitive if the natural action of its automorphism group on its edge set is transitive. An automorphism of a graph is semiregular if all of the orbits of the subgroup generated by this automorphism have the same length.…

Let $X$ be a finite vertex-transitive graph of valency $d$, and let $A$ be the full automorphism group of $X$. Then the arc-type of $X$ is defined in terms of the sizes of the orbits of the action of the stabiliser $A_v$ of a given vertex…

Let $S, K$ be two subrings of a finite ring $R$. Then the generalized non-commuting graph of subrings $S, K$ of $R$, denoted by $\Gamma_{S, K}$, is a simple graph whose vertex set is $(S \cup K) \setminus (C_K(S) \cup C_S(K))$ and two…

We classify non-complete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of $2$-geodesics. We prove that either $\Gamma$ is 2-arc transitive or the valency $p$…

A finite graph is called a tricirculant if admits a cyclic group of automorphism which has precisely three orbits on the vertex-set of the graph, all of equal size. We classify all finite connected cubic vertex-transitive tricirculants. We…

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group ${\rm Aut}(A_\Gamma)$. In particular, we prove that a finite normal subgroup in ${\rm Aut}(A_\Gamma)$ has at most order…

Circulant graphs are a widely studied family of graphs whose members possess varying amounts of symmetry. Although considerable progress has been made in finding the automorphism groups of circulant graphs under certain restrictions, a…

It is well-known that a simple $G$-arc-transitive graph can be represented as a coset graph for the group $G$. This representation is extended to a construction of $G$-arc-transitive coset graphs $\Cos(G,H,J)$ with finite valency and finite…

A detailed description of the structure of two-ended arc-transitive digraphs is given. It is also shown that several sets of conditions, involving such concepts as Property Z, local quasi-primitivity and prime out-valency, imply that an…

A graph is called {\em arc-transitive} (or {\em symmetric}) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In…

We introduce the notion of an \emph{$n$-dimensional mixed dihedral group}, a general class of groups for which we give a graph theoretic characterisation. In particular, if $H$ is an $n$-dimensional mixed dihedral group then the we…

A graph $\Gamma$ is said to be a semi-Cayley graph over a group $G$ if it admits $G$ as a semiregular automorphism group with two orbits of equal size. We say that $\Gamma$ is normal if $G$ is a normal subgroup of ${\rm Aut}(\Gamma)$. We…

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 $\Gamma$…

Let $R$ be a noncommutative ring with identity. The commuting graph of $R$, denoted by $\Gamma(R)$, is a graph with vertex set $R \setminus Z(R)$, and two vertices $a$, $b$ are adjacent if $a\neq b$ and $ab=ba$. Let $T=Tr(R)$ be the ring of…

A graph $\Gamma$ is called edge-regular whenever it is regular and for any two adjacent vertices, the number of their common neighbors is independent of the choice of vertices. A clique $C$ in $\Gamma$ is called regular whenever for any…

The main result of this paper is that, if $\Gamma$ is a connected 4-valent $G$-arc-transitive graph and $v$ is a vertex of $\Gamma$, then either $\Gamma$ is one of a well understood infinite family of graphs, or $|G_v|\leq 2^43^6$ or…

In this paper we consider connected locally $G$-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighourhood $\Gamma(u) \cup \Gamma(v)$ is trivial. We…

Let $S$ be a subring of a finite ring $R$ and $C_R(S) = \{r \in R : rs = sr \;\forall\; s \in S\}$. The relative non-commuting graph of the subring $S$ in $R$, denoted by $\Gamma_{S, R}$, is a simple undirected graph whose vertex set is $R…

This paper presents a solution of the polycirculant conjecture which states that every vertex-transitive graph G has an automorphism that permutes the vertices in cycles of the same length. This is done by identifying vertex-transitive…

We prove that, if $\Gamma$ is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of $\Gamma$ of order at least $6$, or the number of vertices of $\Gamma$ is bounded above by an absolute…