arXiv subjects
Publications and source records attributed to Yan-Quan Feng.
With help of group characters we enumerate orientably-regular maps of a given type with automorphism group isomorphic to $\PGam(2,2^p)$ for prime $p$.
A Cayley (di)graph $\Cay(G,S)$ of a finite group $G$ is called CI if, for every Cayley (di)graph $\Cay(G,T)$ of $G$, $\Cay(G,S)\cong \Cay(G,T)$ implies that $S^{\sigma}=T$ for some $\sigma\in \Aut(G)$. The group $G$ is called an NDCI-group (resp. NCI-group) if every normal Cayley digraph (resp. graph) of $G$ is CI. It was shown that the generalized quaternion group $\Q_{4n}$ of order $4n$ ($n\geq 2$) is an NDCI-group if and only if either $n=2$ or $n$ is odd, but its NCI-group classification has been left as an open question. In this paper, we solve the question and prove that $\Q_{4n}$ is an NCI-group for every $n\geq 2$. A normal Cayley (di)graph of a group $G$ is called NNN if its automorphism group contains a non-normal regular subgroup isomorphic to $G$, and $G$ is called an NNND-group (resp. NNN-group) if it admits an NNN Cayley digraph (resp. graph). In this paper, we show that $\Q_{4n}$ is not an NNN-group for every $n\geq 2$, and is an NNND-group if and only if $n\geq 6$ and $n$ is even.
The present work investigates regular, semiregular, and chiral polytopes of any rank $d\geq 3$, whose automorphism groups are 2-groups. There is a large variety of rather small finite regular or alternating semiregular polytopes with automorphism groups of 2-power order: for such polytopes with toroidal sections of rank 3, the various sections of rank 3 can be entirely prescribed (possibly with one exception in the semiregular case). It is also shown that having a 2-group as automorphism group is hereditary under taking universal extensions: the universal extension of a given regular, chiral, or alternating semiregular polytope with a finite or infinite 2-group as automorphism group, is a polytope of one rank higher with an infinite 2-group an automorphism group.
Let $\mathcal{P}$ be a chiral polytope with type $\{k_1, k_2\}$ and $G=Aut(\mathcal{P})$. Suppose $|G|=2p^m$, where $k_1, k_2\geq 3$ and $p$ is an odd prime. Let $P$ be a Sylow $p$-subgroup of $G$. We prove that $G \cong P \rtimes \mathbb{Z}_2$, $d(P)=2$, $P' \ne 1$(so $m \geq 3$) and up to duality, $\{k_1, k_2\}=\{p^{l_1}, 2p^{l_2}\}$ for some integral $l_1, l_2 \geq 1$. Moreover, we show that $\mathcal{P}$ is tight $(k_1k_2=2p^m$) if and only if $P$ is metacyclic group. Furthermore, if $m=3$ or $4$, then $\mathcal{P}$ must be tight, and if $m \geq 5$, where either $m$ is odd, or $m$ is even and $m \geq p+3$, there exists a non-tight chiral polytope $\mathcal{P}$.
The isomorphism problem for digraphs is a fundamental problem in graph theory. In this paper, we consider this problem for $m$-Cayley digraphs which are generalization of Cayley digraphs. Let $m$ be a positive integer. A digraph admitting a group $G$ of automorphisms acting semiregularly on the vertices with exactly $m$ orbits is called an $m$-Cayley digraph of $G$. In our previous paper, we developed a theory for $m$-Cayley isomorphisms of $m$-Cayley digraphs, and classified finite $m$CI-groups for each $m\geq 2$, and finite $m$PCI-groups for each $m\geq 4$. The next natural step is to classify finite $m$PCI-groups for $m=2$ or $3$. Note that BCI-groups form an important subclass of the $2$PCI-groups, which were introduced in 2008 by Xu et al. Despite much effort having been made on the study of BCI-groups, the problem of classifying finite BCI-groups is still widely open. In this paper, we prove that every finite $2$PCI-group is solvable, and its Sylow $3$-subgroup is isomorphic to $Z_3, Z_3\times Z_3$ or $Z_9$, and Sylow $p$-subgroup with $p\not=3$ is either elementary abelian, or isomorphic to $Z_4$ or $Q_8$. We also introduce the kernel isomorphisms of $m$-Cayley digraphs, and establish some useful theory for studying this kind of isomorphisms. Using the results of kernel isomorphisms of $m$-Cayley digraphs together with the results on $2$PCI-groups, we give a proper description of finite BCI-groups, and in particular, we obtain a complete classification of finite non-abelian BCI-groups.
A Cayley digraph on a group $G$ is called NNN if the Cayley digraph is normal and its automorphism group contains a non-normal regular subgroup isomorphic to $G$. A group is called NNND-group or NNN-group if there is an NNN Cayley digraph or graph on the group, respectively. In this paper, it is shown that there is no cyclic NNND-group, and hence no cyclic NNN-group. Furthermore, a dihedral group of order $2n$ is an NNND-group or an NNN-group if and only if $n\ge 6$ is even and $n\not=8$.
Let $\Gamma$ be a connected $7$-valent symmetric Cayley graph on a finite non-abelian simple group $G$. If $\Gamma$ is not normal, Li {\em et al.} [On 7-valent symmetric Cayley graphs of finite simple groups, J. Algebraic Combin. 56 (2022) 1097-1118] characterised the group pairs $(\mathrm{soc}(\mathrm{Aut}(\Gamma)/K),GK/K)$, where $K$ is a maximal intransitive normal subgroup of $\mathrm{Aut}(\Gamma)$. In this paper, we improve this result by proving that if $\Gamma$ is not normal, then $\mathrm{Aut}(\Gamma)$ contains an arc-transitive non-abelian simple normal subgroup $T$ such that $G<T$ and $(T,G)=(\mathrm{A}_{n},\mathrm{A}_{n-1})$ with $n=7$, $3\cdot 7$, $3^2\cdot 7$, $2^2\cdot 3\cdot 7$, $2^3\cdot3\cdot7$, $2^3\cdot3^2\cdot5\cdot7$, $2^4\cdot3^2\cdot5\cdot7$, $2^6\cdot3\cdot7$, $2^7\cdot3\cdot7$, $2^6\cdot3^2\cdot7$, $2^6\cdot3^4\cdot5^2\cdot7$, $2^8\cdot3^4\cdot5^2\cdot7$, $2^7\cdot3^4\cdot5^2\cdot7$, $2^{10}\cdot3^2\cdot7$, $2^{24}\cdot3^2\cdot7$. Furthermore, $\mathrm{soc}(\mathrm{Aut}(\Gamma)/R)=(T\times R)/R$, where $R$ is the largest solvable normal subgroup of $\mathrm{Aut}(\Gamma)$.
Extending the well-studied concept of graphical regular representations to bipartite graphs, a Haar graphical representation (HGR) of a group $G$ is a bipartite graph whose automorphism group is isomorphic to $G$ and acts semiregularly with the orbits giving the bipartition. The question of which groups admit an HGR was inspired by a closely related question of Est\'elyi and Pisanski in 2016, as well as Babai's work in 1980 on poset representations, and has been recently solved by Morris and Spiga. In this paper, we introduce the $m$-Haar graphical representation ($m$-HGR) as a natural generalization of HGR to $m$-partite graphs for $m\geq2$, and explore the existence of $m$-HGRs for any fixed group. This inquiry represents a more robust version of the existence problem of G$m$SRs as addressed by Du, Feng and Spiga in 2020. Our main result is a complete classification of finite groups $G$ without $m$-HGRs.
Let $m$ be a positive integer. A group $G$ is said to be an $m$-DCI-group or an $m$-CI-group if $G$ has the $k$-DCI property or $k$-CI property for all positive integers $k$ at most $m$, respectively. Let $G$ be a dihedral group of order $2n$ with $n\geq 3$. Qu and Yu proved that $G$ is an $m$-DCI-group or $m$-CI-group, for every $m\in \{1,2,3\}$, if and only if $n$ is odd. In this paper, it is shown that $G$ is a $4$-DCI-group if and only if $n$ is odd and not divisible by $9$, and $G$ is a $4$-CI-group if and only if $n$ is odd.
Let $(G,\{\rho_0, \rho_1, \rho_2\})$ be a string C-group of order $4p^m$ with type $\{k_1, k_2\}$ for $m \geq 2$, $k_1, k_2\geq 3$ and $p$ be an odd prime. Let $P$ be a Sylow $p$-subgroup of $G$. We prove that $G \cong P \rtimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$, $d(P)=2$, and up to duality, $p \mid k_1, 2p \mid k_2$. Moreover, we show that if $P$ is abelian, then $(G,\{\rho_0, \rho_1, \rho_2\})$ is tight and hence known. In the case where $P$ is nonabelian, we construct an infinite family of string C-group with type $\{p, 2p\}$ of order $4p^m$ where $m \geq 3$.
A vertex transitive graph $Γ$ is said to be $2$-distance transitive if for each vertex $u$, the group of automorphisms of $Γ$ fixing the vertex $u$ acts transitively on the set of vertices at distance $1$ and $2$ from $u$, while $Γ$ is said to be $2$-arc transitive if its automorphism group is transitive on the set of $2$-arcs. Then $2$-arc transitive graphs are $2$-distance transitive. The classification of $2$-arc transitive Cayley graphs on dihedral groups was given by Du, Malnič and Marušič in [Classification of 2-arc-transitive dihedrants, J. Combin. Theory Ser. B 98 (2008), 1349--1372]. In this paper, it is shown that a connected 2-distance transitive Cayley graph on the dihedral group of order $2n$ is either $2$-arc transitive, or isomorphic to the complete multipartite graph $K_{m[b]}$ for some $m\geq3$ and $b\geq2$ with $mb=2n$.
A Cayley digraph Cay(G,S) of a finite group $G$ with respect to a subset $S$ of $G$ is said to be a CI-digraph if for every Cayley digraph Cay(G,T) isomorphic to Cay(G,S), there exists an automorphism $σ$ of $G$ such that $S^σ=T$. A finite group $G$ is said to have the $m$-DCI property for some positive integer $m$ if all $m$-valent Cayley digraphs of $G$ are CI-digraphs, and is said to be a DCI-group if $G$ has the $m$-DCI property for all $1\leq m\leq |G|$. Let $\mathrm{Q}_{4n}$ be a generalized quaternion group of order $4n$ with an integer $n\geq 3$, and let $\mathrm{Q}_{4n}$ have the $m$-DCI property for some $1 \leq m\leq 2n-1$. It is shown in this paper that $n$ is odd, and $n$ is not divisible by $p^2$ for any prime $p\leq m-1$. Furthermore, if $n\geq 3$ is a power of a prime $p$, then $\mathrm{Q}_{4n}$ has the $m$-DCI property if and only if $p$ is odd, and either $n=p$ or $1\leq m\leq p$.
The spined cube $SQ_n$ is a variant of the hypercube $Q_n$, introduced by Zhou et al. in [Information Processing Letters 111 (2011) 561-567] as an interconnection network for parallel computing. A graph $\G$ is an $m$-Cayley graph if its automorphism group $\Aut(\G)$ has a semiregular subgroup acting on the vertex set with $m$ orbits, and is a Caley graph if it is a 1-Cayley graph. It is well-known that $Q_n$ is a Cayley graph of an elementary abelian 2-group $\mz_2^n$ of order $2^n$. In this paper, we prove that $SQ_n$ is a 4-Cayley graph of $\mz_2^{n-2}$ when $n\geq6$, and is a $\lfloor n/2\rfloor$-Cayley graph when $n\leq 5$. This symmetric property shows that an $n$-dimensional spined cube with $n\geq6$ can be decomposed to eight vertex-disjoint $(n-3)$-dimensional hypercubes, and as an application, it is proved that there exist two edge-disjoint Hamiltonian cycles in $SQ_n$ when $n\geq4$. Moreover, we determine the vertex-transitivity of $SQ_n$, and prove that $SQ_n$ is not vertex-transitive unless $n\leq3$.
A Cayley digraph $\rm{Cay}(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called a CI-digraph if for any Cayley digraph $\rm{Cay}(G,T)$ isomorphic to $\rm{Cay}(G,S)$, there is an $α\in \rm{Aut}(G)$ such that $S^α=T$. For a positive integer $m$, $G$ is said to have the $m$-DCI property if all Cayley digraphs of $G$ with out-valency $m$ are CI-digraphs. Li [The Cyclic groups with the $m$-DCI Property, European J. Combin. 18 (1997) 655-665] characterized cyclic groups with the $m$-DCI property, and in this paper, we characterize dihedral groups with the $m$-DCI property. For a dihedral group $\mathrm{D}_{2n}$ of order $2n$, assume that $\mathrm{D}_{2n}$ has the $m$-DCI property for some $1 \leq m\leq n-1$. Then it is shown that $n$ is odd, and if further $p+1\leq m\leq n-1$ for an odd prime divisor $p$ of $n$, then $p^2\nmid n$. Furthermore, if $n$ is a power of a prime $q$, then $\mathrm{D}_{2n}$ has the $m$-DCI property if and only if either $n=q$, or $q$ is odd and $1\leq m\leq q$.
A finite group $G$ admits an {\em oriented regular representation} if there exists a Cayley digraph of $G$ such that it has no digons and its automorphism group is isomorphic to $G$. Let $m$ be a positive integer. In this paper, we extend the notion of oriented regular representations to oriented $m$-semiregular representations using $m$-Cayley digraphs. Given a finite group $G$, an {\em $m$-Cayley digraph} of $G$ is a digraph that has a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. We say that a finite group $G$ admits an {\em oriented $m$-semiregular representation} if there exists a regular $m$-Cayley digraph of $G$ such that it has no digons and $G$ is isomorphic to its automorphism group. In this paper, we classify finite groups admitting an oriented $m$-semiregular representation for each positive integer $m$.
A vertex triple $(u,v,w)$ of a graph is called a $2$-geodesic if $v$ is adjacent to both $u$ and $w$ and $u$ is not adjacent to $w$. A graph is said to be $2$-geodesic transitive if its automorphism group is transitive on the set of $2$-geodesics. In this paper, a complete classification of $2$-geodesic transitive graphs of order $p^n$ is given for each prime $p$ and $n\leq 3$. It turns out that all such graphs consist of three small graphs: the complete bipartite graph $K_{4,4}$ of order $8$, the Schläfli graph of order $27$ and its complement, and fourteen infinite families: the cycles $C_p, C_{p^2}$ and $C_{p^3}$, the complete graphs $K_p, K_{p^2}$ and $K_{p^3}$, the complete multipartite graphs $K_{p[p]}$, $K_{p[p^2]}$ and $K_{p^2[p]}$, the Hamming graph $H(2,p)$ and its complement, the Hamming graph $H(3,p)$, and two infinite families of normal Cayley graphs on extraspecial group of order $p^3$ and exponent $p$.
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, Malnič, Martínez and Marušič in 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 $p\geq 5$ be a prime and $Γ$ a connected symmetric graph of valency $p$ admitting a quasi-semiregular automorphism. In this paper, we first prove that either $Γ$ 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 $Γ$ 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 $Γ$ such that either $\rm{Aut}(Γ)$ has a solvable arc-transitive subgroup or $Γ$ 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 group $G$ admits an \textbf{\em $n$-partite digraphical representation} if there exists a regular $n$-partite digraph $Γ$ such that the automorphism group $\mathrm{Aut}(Γ)$ of $Γ$ satisfies the following properties: $\mathrm{Aut}(Γ)$ is isomorphic to $G$, $\mathrm{Aut}(Γ)$ acts semiregularly on the vertices of $Γ$ and the orbits of $\mathrm{Aut}(Γ)$ on the vertex set of $Γ$ form a partition into $n$ parts giving a structure of $n$-partite digraph to $Γ$. In this paper, for every positive integer $n$, we classify the finite groups admitting an $n$-partite digraphical representation.