Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group $\text{A}_{39}$. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group $\text{A}_{40}$.
A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.
Abstract
A Cayley graph Γ is said to be arc-transitive if its full automorphism group AutΓ is transitive on the arc set of Γ. In this paper we give a characterization of pentavalent arc-transitive Cayley graphs on a class of Frobenius groups with soluble vertex stabilizer.
A Cayley graph Cay (G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay (G,S). For a positive integer n, let Γn be a graph with vertex set {xi,yi|i ∈ ℤ2n} and edge set {{xi,xi+1}, {yi,yi+1}, {x2i,y2i+1}, {y2i,x2i+1}|i ∈ ℤ2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to [Formula: see text] for n=2, and to [Formula: see text] for n > 2. Furthermore, we determine all pairs of G and S such that Γn= Cay (G,S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.
A graph is said to be a bi-Cayley graph over a group $H$ if it admits $H$ as a group of automorphisms acting semiregularly on its vertices with two orbits. For a prime $p$, we call a bi-Cayley graph over a metacyclic $p$-group a bi-$p$-metacirculant. In this paper, the automorphism group of a connected cubic edge-transitive bi-$p$-metacirculant is characterized for an odd prime $p$, and the result reveals that a connected cubic edge-transitive bi-$p$-metacirculant exists only when $p=3$. Using this, a classification is given of connected cubic edge-transitive bi-Cayley graphs over an inner-abelian metacyclic $3$-group. As a result, we construct the first known infinite family of cubic semisymmetric graphs of order twice a $3$-power.
A Cayley graph ${\rm Cay}(G,S)$ on a group $G$ is said to be normal if the right regular representation $R(G)$ of $G$ is normal in the full automorphism group of ${\rm Cay}(G,S)$. In this paper, all connected tetravalent non-normal Cayley graphs of order $4p$ are constructed explicitly for each prime $p$. As a result, there are fifteen sporadic and eleven infinite families of tetravalent non-normal Cayley graphs of order $4p$.
AbstractAn automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.
Let Sndenote the symmetric group of degree n with n ≥ 3, S = { cn= (1 2 ⋯ n), [Formula: see text], (1 2)} and Γn= Cay(Sn, S) be the Cayley graph on Snwith respect to S. In this paper, we show that Γn(n ≥ 13) is a normal Cayley graph, and that the full automorphism group of Γnis equal to Aut(Γn) = R(Sn) ⋊ 〈Inn(ϕ) ≅ Sn× ℤ2, where R(Sn) is the right regular representation of Sn, ϕ = (1 2)(3 n)(4 n−1)(5 n−2) ⋯ (∊ Sn), and Inn(ϕ) is the inner isomorphism of Sninduced by ϕ.
We are concerned here with question: to what extent can the structure of a group G be recaptured from information about the structure of its group of automorphismsAut G? For example, one might try to find all groups which have some specific group astheir (full) automorphism group, a point of view adopted by Iyer in a recent paper [5]. Nothing is known about this question in general except the result of Nagrebeckü [7] that there are only finitely many finite groups with a given group as automorphismgroup.
The degree pattern of a finite group G denoted by D(G) was introduced in [5]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and same degree pattern as G. In the present article, we show that the alternating group A10 and the automorphism group Aut (McL) are 2-fold OD-characterizable, while the automorphism group Aut (J2) is 3-fold OD-characterizable and the symmetric group S10 is 8-fold OD-characterizable. It is worth mentioning that the prime graphs associated to these groups are connected and, in fact, among the groups with this property, they are the first groups which are investigated for OD-characterizability.
A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A119