Prime Valency Research Articles

On 2-distance primitive graphs of prime valency

A vertex transitive non-complete graph is called 2-distance primitive if the stabilizer of a vertex is primitive on both the first and the second step neighbourhoods. In 2019, Jin, Huang and Liu proved that all connected 2-distance primitive graphs of prime valency must belong to some known families of distance-transitive graphs if the socle of the stabilizer of a vertex is not a 2-transitive linear group with some further restrictions. They posed a problem to determine the connected 2-distance primitive graphs of prime valency when the socle of the stabilizer of a vertex is a 2-transitive linear group. In this paper, we make a crucial progress to this problem by proving that either the graph under consideration in this problem is the folded 5-cube or the socle of the stabilizer of a vertex is PSL(2,q) and the graph has girth 4 and contains no 5-cycles. We also give a short proof of the main result of Jin, Huang and Liu (2019), and we find some new graphs that are not in the list of Jin, Huang and Liu (2019).

On Symmetric bi-Cayley Graphs of Prime Valency on Nonabelian Simple Groups

Let $\Gamma$ be a bipartite graph, and let $\mathrm{Aut}\Gamma$ be the full automorphism group of the graph $\Gamma$. A subgroup $G\leqslant \mathrm{Aut}\Gamma$ is said to be bi-regular on $\Gamma$ if $G$ preserves the bipartition and acts regularly on both parts of $\Gamma$, while the graph $\Gamma$ is called a bi-Cayley graph of $G$ in this case. A subgroup $X\leqslant \mathrm{Aut} \Gamma$ is said to be bi-quasiprimitive on $\Gamma$ if the bipartition-preserving subgroup of $X$ is a quasiprimitive group on each part of $\Gamma$.
 In this paper, a characterization is given for the connected bi-Cayley graphs on nonabelian simple groups which have prime valency and admit bi-quasiprimitive groups.

Symmetric bi-Cayley graphs on nonabelian simple groups with prime valency

Yin et al. (2021) [19] classified connected arc-transitive Cayley graphs on nonabelian simple groups with prime valency p≥11 and solvable vertex stabilizers. In this paper, we extend this result to bi-Cayley graphs. Let Γ be a connected arc-transitive bi-Cayley graph on a nonabelian simple group T with prime valency p≥5 and solvable vertex stabilizer. It is proved that either Γ is nonbipartite, or Γ is a normal bi-Cayley graph, or Γ is S-semisymmetric with (S,T,p)=(PSL2(11),A5,11),(PSL2(29),A5,29),(M23,M22,23), or (An,An−1,p) where p|n|pkl and k|l|(p−1). Moreover, example exists for each triple (S,T,p) above.

Semisymmetric graphs of order twice prime powers with the same prime valency

Semisymmetric graphs of order twice prime powers with the same prime valency

Prime-valent symmetric graphs with a quasi-semiregular automorphism

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 problem regarding existence of semiregular automorphisms in vertex-transitive graphs. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers (Feng et al., 2019 [11]) and (Yin and Feng, 2021 [42]).Let Γ be a connected symmetric graph of prime valency p≥5 admitting a quasi-semiregular automorphism. In this paper, it is proved that either Γ is a connected Cayley graph 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. Further, for p=5 a complete classification of graphs Γ such that either Aut(Γ) has a solvable arc-transitive subgroup or Γ is T-arc-transitive with T a non-abelian simple group is given. Finally, a construction of an infinite family of symmetric graphs admitting a quasi-semiregular automorphism and having nonsolvable automorphism group is given.

2-Arc-transitive Cayley graphs on alternating groups

An interesting fact is that almost all the connected 2-arc-transitive nonnormal Cayley graphs on nonabelian simple groups with small valency or prime valency (provided solvable vertex stabilizers) are Cayley graphs on alternating groups An. This naturally motivates the study of 2-arc-transitive Cayley graphs on An for arbitrary valency. In this paper, we characterize the automorphism groups of such graphs. In particular, we show that for a non-complete (G,2)-arc-transitive Cayley graph on An with G almost simple, the socle of G is either An+1 or An+2. We also construct the first infinite family of (An+2,2)-arc-transitive Cayley graphs on An.

Symmetric graphs of prime valency associated with some almost simple groups

A graph is symmetric if its automorphism group acts transitively on both the vertex and the arc sets. Stimulated by Lorimer's work on symmetric graphs of prime valency, we make a further investigation on the structural properties of the automorphism groups of connected symmetric graphs with prime valency. Also, as an application, we give a classification for symmetric graphs of prime valency arising from a class of almost simple groups.

Arc-transitive Cayley graphs on nonabelian simple groups with prime valency

In 2011, Fang et al. in [9] posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency d, where eitherd≤20or 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 valencyd≥4. Even for this problem, it was only solved for the cases when either d≤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 d≥11 is a prime and the vertex stabilizer is solvable.

On quasiprimitive edge-transitive graphs of odd order and twice prime valency

Abstract A graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let Γ {\varGamma} be a connected graph of odd order and twice prime valency, and let G be a subgroup of the automorphism group of Γ {\varGamma} . In the case where G acts transitively on the edge set and quasiprimitively on the vertex set of Γ {\varGamma} , we prove that either G is almost simple, or G is a primitive group of affine type. If further G is an almost simple primitive group, then, with two exceptions, the socle of G acts transitively on the edge set of Γ {\varGamma} .

阶为8p<sup>2</sup>的7度对称图

阶为8p<sup>2</sup>的7度对称图

S-Arc-regular prime-valent Cayley graphs of square-free order

A high cited paper [Feng and Li, One-regular graphs of square-free order of prime valency, Europ. J. Combin. 32 (2011) 261–275] classified 1-arc-regular prime-valent graphs (necessarily Cayley graphs) of square-free order. It is thus interesting to ask what are the [Formula: see text]-arc-regular prime-valent Cayley graphs of square-free order for each [Formula: see text]. In this paper, a complete list of all such graphs is given. The proving process also involves a complete list of the vertex stabilizers of [Formula: see text]-arc-regular prime-valent graphs of any order.

Two-distance-primitive graphs with prime valency

A 2-distance-primitive graph is a vertex-transitive graph satisfying that the stabilizer of a vertex is primitive on both the first step and the second step neighbourhoods. In this paper, we give a classification of the family of prime valency 2-distance-primitive graphs.

Regular self-dual and self-Petrie-dual maps of arbitrary valency

The existence of a regular, self-dual and self-Petrie-dual map of any given even valency has been proved by D. Archdeacon, M. Conder and J. Siraň (2014). In this paper we extend this result to any odd valency ≥ 5 . This is done using algebraic number theory and maps defined on the groups PSL(2, p ) in the case of odd prime valency ≥ 5 and valency 9 , and using coverings for the remaining odd valencies.

On Arc-Transitive Metacyclic Covers of Graphs with Order Twice a Prime

Quite a lot of attention has been paid recently to the characterization and construction of edge- or arc-transitive abelian (mostly cyclic or elementary abelian) covers of symmetric graphs, but there are rare results for nonabelian covers since the voltage graph techniques are generally not easy to be used in this case. In this paper, we will classify certain metacyclic arc-transitive covers of all non-complete symmetric graphs with prime valency and twice a prime order $2p$ (involving the complete bipartite graph ${\sf K}_{p,p}$, the Petersen graph, the Heawood graph, the Hadamard design on $22$ points and an infinite family of prime-valent arc-regular graphs of dihedral groups). A few previous results are extended.

Finite edge-primitive graphs of prime valency

Many famous graphs are edge-primitive. Weiss (1973) and Guo et al. (2013) determined edge-primitive graphs of valency 3 and 5, respectively. In this paper, we study edge-primitive graphs of any prime valency. It is proved that all such graphs are 2-arc-transitive and the full automorphism groups are almost simple with the only exception the graphs being the complete bipartite graphs, and a complete classification is given of such graphs with soluble edge stabilizers, which widely extends the classifications of Weiss and Guo et al. (notice that the edge stabilizers of edge-primitive graphs with valency at most 5 are soluble).

A characterisation on arc-transitive graphs of prime valency

Let Γ be a finite simple undirected graph and G ≤ Aut(Γ). If G is transitive on the set of s-arcs but not on the set of (s+1)-arcs of Γ, then Γ is called (G, s)-transitive. For a connected (G, s)-transitive graph Γ of prime valency, the vertex-stabilizer Gα with α ∈ V(Γ) has been determined when Gα is solvable. In this paper, we give a characterization of the vertex-stabilizers of (G, s)-transitive graphs of prime valency when Gα is unsolvable.

平方自由阶素数度2-弧正则图

平方自由阶素数度2-弧正则图

On prime-valent symmetric graphs of square-free order

Symmetric graphs of valencies 3, 4 and 5 and square-free order have been classified in the literature. In this paper, we will present a complete classification of symmetric graphs of square-free order and any prime valency which admit a soluble arc-transitive group, and a complete classification of 7-valent symmetric graphs of square-free order.

Permutation groups and derangements of odd prime order

Let G be a transitive permutation group of degree n. We say that G is 2′-elusive if n is divisible by an odd prime, but G does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive 2′-elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate automorphisms of finite arc-transitive graphs of prime valency.

Arc-transitive cyclic covers of graphs with order twice a prime

Arc-transitive cyclic covers of symmetric graphs with a specific prime valency and order twice a prime have been studied by nearly ten papers in the literature. In this paper, we will give a general characterization of arc-transitive cyclic covers of graphs with any prime valency and order twice a prime.