Related papers: On basic graphs of symmetric graphs of valency fiv…

A connected symmetric graph of prime valency is {\em basic} if its automorphism group contains no nontrivial normal subgroup having more than two orbits. Let $p$ be a prime and $n$ a positive integer. In this paper, we investigate…

A graph $\Gamma$ is basic if Aut$\Gamma$ has no normal subgroup $N\ne1$ such that $\Gamma$ is a normal cover of the normal quotient graph $\Gamma_N$. In this paper, we completely determine the basic normal quotient graphs of all connected…

A graph \Gamma is said to be {\em symmetric} if its automorphism group \Aut(\Gamma) is transitive on the arc set of \Gamma. Let $G$ be a finite non-abelian simple group and let \Gamma be a connected pentavalent symmetric graph such that…

A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of…

A graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\Gamma$, where an arc is an ordered pair of adjacent vertices. Let $\Gamma$ be a $G$-symmetric graph such that its…

A graph $\Ga=(V,E)$ is called a Cayley graph of some group $T$ if the automorphism group $\Aut(\Ga)$ contains a subgroup $T$ which acts on regularly on $V$. If the subgroup $T$ is normal in $\Aut(\Ga)$ then $\Ga$ is called a normal Cayley…

A graph is called {\em half-arc-transitive} if its full automorphism group acts transitively on vertices and edges, but not on arcs. It is well known that for any prime $p$ there is no tetravalent half-arc-transitive graph of order $p$ or…

A graph $\Gamma$ is said to be symmetric if its automorphism group $\rm Aut(\Gamma)$ acts transitively on the arc set of $\Gamma$. In this paper, we show that if $\Gamma$ is a finite connected heptavalent symmetric graph with solvable…

A connected graph $\Gamma=(V,E)$ of valency at least $3$ is called a basic $2$-arc-transitive graph if its full automorphism group has a subgroup $G$ with the following properties: (i) $G$ acts transitively on the set of $2$-arcs of…

A graph $\Ga$ is $G$-symmetric if $\Ga$ admits $G$ as a group of automorphisms acting transitively on the set of vertices and the set of arcs of $\Ga$, where an arc is an ordered pair of adjacent vertices. In the case when $G$ is…

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…

A graph $\Gamma$ is $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. If $V(\Gamma)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such…

An interesting fact is that most of the known connected $2$-arc-transitive nonnormal Cayley graphs of small valency on finite simple groups are $(\mathrm{A}_{n+1},2)$-arc-transitive Cayley graphs on $\mathrm{A}_n$. This motivates the study…

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malni\v{c}, Mart\'{i}nez and…

A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper,…

Let $\Gamma$ be a graph and let $G$ be a group of automorphisms of $\Gamma$. The graph $\Gamma$ is called $G$-normal if $G$ is normal in the automorphism group of $\Gamma$. Let $T$ be a finite non-abelian simple group and let $G = T^l$ with…

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…

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…

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…

A graph $\Gamma$ is called $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of $G$-symmetric graphs $\Gamma$ with $V(\Gamma)$ admitting…