arXiv subjects

Yan-Quan Feng

Publications and source records attributed to Yan-Quan Feng.

50 records · Page 3Linked to original sources

arXiv2017

On basic graphs of symmetric graphs of valency five

A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the quotient graph $\G_N$ has the same valency with $\G$. In this paper, we classify symmetric basic graphs of order $2qp^n$ and valency 5, where $q<p$ are two primes and $n$ is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order $2q$ with $5\di (q-1)$, the complete graph $K_6$ of order $6$, the complete bipartite graph $K_{5,5}$ of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order $kp^n$ for some small integers $k$ and $n$ are classified.

math.CO↗

arXiv2017

On groups all of whose Haar graphs are Cayley graphs

A Cayley graph of a group $H$ is a finite simple graph $Γ$ such that ${\rm Aut}(Γ)$ contains a subgroup isomorphic to $H$ acting regularly on $V(Γ)$, while a Haar graph of $H$ is a finite simple bipartite graph $Σ$ such that ${\rm Aut}(Σ)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(Σ)$ and the $H$-orbits are equal to the bipartite sets of $Σ$. 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.

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arXiv2017

Bipartite bi-Cayley graphs over metacyclic groups of odd prime-power order

A graph $Γ$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $Γ$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $Γ$ be a connected bipartite bi-Cayley graph over the group $G$. In this paper, we prove that $G$ is normal in the full automorphism group ${\rm Aut}(Γ)$ of $Γ$ when $G$ is a Sylow $p$-subgroup of ${\rm Aut}(Γ)$. As an application, we classify half-arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $2p$. Furthermore, it is shown that there are no semisymmetric and no arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $p$.

math.CO↗

arXiv2017

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.

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arXiv2017

Elementary abelian groups of rank 5 are DCI-groups

In this paper, we show that the group $\mathbb{Z}_p^5$ is a DCI-group for any odd prime $p,$ that is, two Cayley digraphs Cay$(\mathbb{Z}_p^5,S)$ and Cay$(\mathbb{Z}_p^5,T)$ are isomorphic if and only if $S=T^φ$ for some automorphism $φ$ of the group $\mathbb{Z}_p^5$.

math.CO↗

arXiv2017

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.

math.CO↗

arXiv2017

Equal relation between the extra connectivity and pessimistic diagnosability for some regular graphs

Extra connectivity and the pessimistic diagnosis are two crucial subjects for a multiprocessor system's ability to tolerate and diagnose faulty processor. The pessimistic diagnosis strategy is a classic strategy based on the PMC model in which isolates all faulty vertices within a set containing at most one fault-free vertex. In this paper, the result that the pessimistic diagnosability $t_p(G)$ equals the extra connectivity $κ_{1}(G)$ of a regular graph $G$ under some conditions are shown. Furthermore, the following new results are gotten: the pessimistic diagnosability $t_p(S_n^2)=4n-9$ for split-star networks $S_n^2$, $t_p(Γ_n)=2n-4$ for Cayley graphs generated by transposition trees $Γ_n$, $t_p(Γ_{n}(Δ))=4n-11$ for Cayley graph generated by the $2$-tree $Γ_{n}(Δ)$, $t_{p}(BP_n)=2n-2$ for the burnt pancake networks $BP_n$. As corollaries, the known results about the extra connectivity and the pessimistic diagnosability of many famous networks including the alternating group graphs, the alternating group networks, BC networks, the $k$-ary $n$-cube networks etc. are obtained directly.

math.CO↗

arXiv2017

Pentavalent symmetric graphs admitting vertex-transitive non-abelian simple groups

A graph Γis said to be {\em symmetric} if its automorphism group \Aut(Γ) is transitive on the arc set of Γ. Let $G$ be a finite non-abelian simple group and let Γbe a connected pentavalent symmetric graph such that G\leq \Aut(Γ). In this paper, we show that if $G$ is transitive on the vertex set of Γ, then either G\unlhd \Aut(Γ) or \Aut(Γ) contains a non-abelian simple normal subgroup $T$ such that $G\leq T$ and $(G,T)$ is one of $58$ possible pairs of non-abelian simple groups. In particular, if $G$ is arc-transitive, then $(G,T)$ is one of $17$ possible pairs, and if $G$ is regular on the vertex set of Γ, then $(G,T)$ is one of $13$ possible pairs, which improves the result on pentavalent symmetric Cayley graph given by Fang, Ma and Wang in 2011.

math.GR↗

arXiv2017

Tetravalent $2$-arc-transitive Cayley graphs on non-abelian simple groups

A graph Gamma is said to be 2-arc-transitive if its full automorphism group Aut(Γ) has a single orbit on ordered paths of length 2, and for G\leq Aut(Γ), Γis G-regular if G is regular on the vertex set of Γ. Let G be a finite non-abelian simple group and let Γbe a connected tetravalent 2-arc-transitive G-regular graph. In 2004, Fang, Li and Xu proved that either G\unlhd \Aut(Γ) or G is one of 22 possible candidates. In this paper, the number of candidates is reduced to 7, and for each candidate G, it is shown that \Aut(Γ) has a normal arc-transitive non-abelian simple subgroup T such that G\leq T and the pair (G,T) is explicitly given

math.GR↗

arXiv2016

On cubic symmetric non-Cayley graphs with solvable automorphism groups

It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a $2$-regular graph of type $2^2$, that is, a graph with no automorphism of order $2$ interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic $2$-regular graphs of type $2^2$ with a solvable automorphism group is constructed. The smallest graph in this family has order 6174.

math.CO↗

arXiv2016

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.

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arXiv2016

Half-arc-transitive graphs of prime-cube order of small valencies

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 $p^2$. Xu~[Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] classified half-arc-transitive graphs of order $p^3$ and valency $4$. In this paper we classify half-arc-transitive graphs of order $p^3$ and valency $6$ or $8$. In particular, the first known infinite family of half-arc-transitive Cayley graphs on non-metacyclic $p$-groups is constructed.

math.CO↗

arXiv2016

Pentavalent symmetric graphs of order twice a prime power

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 properties of connected pentavalent symmetric graphs of order $2p^n$, and it is shown that a connected pentavalent symmetric graph of order $2p^n$ is basic if and only if it is either a graph of order $6$, $16$, $250$, or a graph of three infinite families of Cayley graphs on generalized dihedral groups -- one family has order $2p$ with $p=5$ or $5 \mid (p-1)$, one family has order $2p^2$ with $5 \mid (p\pm 1)$, and the other family has order $2p^4$. Furthermore, the automorphism groups of these basic graphs are computed. Similar works on cubic and tetravalent symmetric graphs of order $2p^n$ have been done. It is shown that basic graphs of connected pentavalent symmetric graphs of order $2p^n$ are symmetric elementary abelian covers of the dipole $\Dip_5$, and with covering techniques, uniqueness and automorphism groups of these basic graphs are determined. Moreover, symmetric $\mz_p^n$-covers of the dipole $\Dip_5$ are classified. As a byproduct, connected pentavalent symmetric graphs of order $2p^2$ are classified.

math.CO↗

arXiv2007

On 2-Fold Covers of Graphs

A regular covering projection $\p\colon \tX \to X$ of connected graphs is $G$-admissible if $G$ lifts along $\p$. Denote by $\tG$ the lifted group, and let $\CT(\p)$ be the group of covering transformations. The projection is called $G$-split whenever the extension $\CT(\p) \to \tG \to G$ splits. In this paper, split 2-covers are considered. Supposing that $G$ is transitive on $X$, a $G$-split cover is said to be $G$-split-transitive if all complements $\bG \cong G$ of $\CT(\p)$ within $\tG$ are transitive on $\tX$; it is said to be $G$-split-sectional whenever for each complement $\bG$ there exists a $\bG$-invariant section of $\p$; and it is called $G$-split-mixed otherwise. It is shown, when $G$ is an arc-transitive group, split-sectional and split-mixed 2-covers lead to canonical double covers. For cubic symmetric graphs split 2-cover are necessarily cannonical double covers when $G$ is 1- or 4-regular. In all other cases, that is, if $G$ is $s$-regular, $s=2,3$ or 5, a necessary and sufficient condition for the existence of a transitive complement $\bG$ is given, and an infinite family of split-transitive 2-covers based on the alternating groups of the form $A_{12k+10}$ is constructed. Finally, chains of consecutive 2-covers, along which an arc-transitive group $G$ has successive lifts, are also considered. It is proved that in such a chain, at most two projections can be split. Further, it is shown that, in the context of cubic symmetric graphs, if exactly two of them are split, then one is split-transitive and the other one is either split-sectional or split-mixed.

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