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, Malniv{c}, Mart{i}nez and Maruv{s}iv{c} i
n 2013, as a generalization of the well-known semiregular automorphism of a graph. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers. Let $pgeq 5$ be a prime and $Gamma$ a connected symmetric graph of valency $p$ admitting a quasi-semiregular automorphism. In this paper, we first prove that either $Gamma$ is a connected Cayley graph $rm{Cay}(M,S)$ such that $M$ is a $2$-group admitting a fixed-point-free automorphism of order $p$ with $S$ as an orbit of involutions, or $Gamma$ is a normal $N$-cover of a $T$-arc-transitive graph of valency $p$ admitting a quasi-semiregular automorphism, where $T$ is a non-abelian simple group and $N$ is a nilpotent group. Then in case $p=5$, we give a complete classification of such graphs $Gamma$ such that either $rm{Aut}(Gamma)$ has a solvable arc-transitive subgroup or $Gamma$ is $T$-arc-transitive with $T$ a non-abelian simple group. We also construct the first infinite family of symmetric graphs that have a quasi-semiregular automorphism and an insolvable full automorphism group.
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $alphain Aut(G)$ such that $S
^alpha=T$, whenever $Cay(G,S)cong Cay(G,T)$ for a Cayley (di)graph $Cay(G,T)$. A finite group $G$ is called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively. Motivated by a conjecture proposed by Adam in 1967, CI-groups and DCI-groups have been actively studied during the last fifty years by many researchers in algebraic graph theory. It takes about thirty years to obtain the classification of cyclic CI-groups and DCI-groups, and recently, the first two authors, among others, classified cyclic NCI-groups and NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCI-groups, their classification is still elusive. In this paper, we prove that a dihedral group of order $2n$ is a NCI-group or a NDCI-group if and only if $n=2,4$ or $n$ is odd. As a direct consequence, we have that if a dihedral group $D_{2n}$ of order $2n$ is a DCI-group then $n=2$ or $n$ is odd-square-free, and that if $D_{2n}$ is a CI-group then $n=2,9$ or $n$ is odd-square-free, throwing some new light on classification of dihedral CI-groups and DCI-groups.
In this paper, we construct an infinite family of normal Cayley graphs, which are $2$-distance-transitive but neither distance-transitive nor $2$-arc-transitive. This answers a question raised by Chen, Jin and Li in 2019 and corrects a claim in a literature given by Pan, Huang and Liu in 2015.
A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if for every
$Tsubseteq G$, $Cay(G,S)cong Cay(G,T)$ implies that there is $sigmain Aut(G)$ such that $S^sigma=T$. We call a group $G$ a NDCI-group if all normal Cayley digraphs of $G$ are CI-digraphs, and a NCI-group if all normal Cayley graphs of $G$ are CI-graphs, respectively. In this paper, we prove that a cyclic group of order $n$ is a NDCI-group if and only if $8 mid n$, and is a NCI-group if and only if either $n=8$ or $8 mid n$.
In 2011, Fang et al. in (J. Combin. Theory A 118 (2011) 1039-1051) posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency $d$, where either $dleq 20$ or $d$ is a prime number. The only case
for which the complete solution of this problem is known is of $d=3$. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valency $dgeq4$. Even for this problem, it was only solved for the cases when either $dleq 5$ or $d=7$ and the vertex stabilizer is solvable. In this paper, we make crucial progress towards the above problems by completely solving the second problem for the case when $dgeq 11$ is a prime and the vertex stabilizer is solvable.
For any positive integers $n, s, t, l$ such that $n geq 10$, $s, t geq 2$, $l geq 1$ and $n geq s+t+l$, a new infinite family of regular 3-hypertopes with type $(2^s, 2^t, 2^l)$ and automorphism group of order $2^n$ is constructed.
In this paper, we consider the possible types of regular maps of order $2^n$, where the order of a regular map is the order of automorphism group of the map. For $n le 11$, M. Conder classified all regular maps of order $2^n$. It is easy to classify
regular maps of order $2^n$ whose valency or covalency is $2$ or $2^{n-1}$. So we assume that $n geq 12$ and $2leq s,tleq n-2$ with $sleq t$ to consider regular maps of order $2^n$ with type ${2^s, 2^t}$. We show that for $s+tleq n$ or for $s+t>n$ with $s=t$, there exists a regular map of order $2^n$ with type ${2^s, 2^t}$, and furthermore, we classify regular maps of order $2^n$ with types ${2^{n-2},2^{n-2}}$ and ${2^{n-3},2^{n-3}}$. We conjecture that, if $s+t>n$ with $s<t$, then there is no regular map of order $2^n$ with type ${2^s, 2^t}$, and we confirm the conjecture for $t=n-2$ and $n-3$.
In this paper we extend the classical notion of digraphical and graphical regular representation of a group and we classify, by means of an explicit description, the finite groups satisfying this generalization. A graph or digraph is called regular i
f each vertex has the same valency, or, the same out-valency and the same in-valency, respectively. An m-(di)graphical regular representation (respectively, m-GRR and m-DRR, for short) of a group G is a regular (di)graph whose automorphism group is isomorphic to G and acts semiregularly on the vertex set with m orbits. When m=1, this definition agrees with the classical notion of GRR and DRR. Finite groups admitting a 1-DRR were classified by Babai in 1980, and the analogue classification of finite groups admitting a 1-GRR was completed by Godsil in 1981. Pivoting on these two results in this paper we classify finite groups admitting an m-GRR or an m-DRR, for arbitrary positive integers m. For instance, we prove that every non-identity finite group admits an m-GRR, for every m>4.
In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schlafli types ${4, 2^{n-3}}, {4, 2^{n-4}}$ and ${4, 2^{n-5}}$ for $n geq 10$, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [P
roblems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53(2006) 231-255].
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 stabilizer admitting
a vertex-transitive non-abelian simple group $G$ of automorphisms, then either $G$ is normal in $rm Aut(Gamma)$, or $rm Aut(Gamma)$ contains a non-abelian simple normal subgroup $T$ such that $Gleq T$ and $(G,T)$ is explicitly given as one of $11$ possible exception pairs of non-abelian simple groups. Furthermore, if $G$ is regular on the vertex set of $Gamma$ then the exception pair $(G,T)$ is one of $7$ possible pairs, and if $G$ is arc-transitive then the exception pair $(G,T)=(A_{17},A_{18})$ or $(A_{35},A_{36})$.
