Automorphism Groups Of Cayley Graphs Research Articles

Automorphism groups of the constituent graphs of integral distance graphs

In this paper, we consider the automorphism groups of Cayley graphs which are a basis of a complete Boolean algebra of strongly regular graphs, one of such graph is the integral distance graph Γ . The automorphism groups of the integral distance graphs Γ were determined by Kovács and Ruff in 2014 and by Kurz in 2009. Our point of interest will be restricted to the one determined by Kovács and Ruff. Here we consider the automorphism groups of subgraphs Γ 1 and Γ 2 , which inherit the properties of Γ . It will be shown that Aut Γ is isomorphic to Aut Γ 2 and that Aut Γ 1 is isomorphic to S F q 2 ≀ S [ 2 ] .

A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A119

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.

Automorphism groups of Cayley graphs of order pq2 where p 6 =/= q are prime numbers

Automorphism groups of Cayley graphs of order pq2 where p 6 =/= q are prime numbers

On Arc-Transitive Pentavalent Cayley Graphs on Finite Nonabelian Simple Groups

A Cayley graph $${\varGamma }=\mathsf{Cay}(G,S)$$ is said to be normal if G is normal in $$\mathsf{Aut}{\varGamma }$$ . The concept of normal Cayley graphs was first proposed by Xu (Discrete Math 182:309–319, 1998) and it plays an important role in determining the full automorphism groups of Cayley graphs. In this paper, we study the normality of connected arc-transitive pentavalent Cayley graphs $${\varGamma }$$ on finite nonabelian simple groups G, where the vertex stabilizer $$\mathsf{A}_v$$ is soluble for $$\mathsf{A}=\mathsf{Aut}{\varGamma }$$ and $$v\in V{\varGamma }$$ . We prove that $${\varGamma }$$ is either normal or $$G=\mathsf{A}_{39}$$ or $$\mathsf{A}_{79}$$ . Further, a connected pentavalent arc-transitive non-normal Cayley graph on $$\mathsf{A}_{79}$$ is constructed. To our knowledge, this is the first known example of pentavalent 3-arc-transitive Cayley graph on finite nonabelian simple group which is non-normal.

The endomorphism monoids and automorphism groups of Cayley graphs of semigroups

In this note, we introduce the notions of color-permutable automorphisms and color-permutable vertex-transitive Cayley graphs of semigroups. As a main result, for a finite monoid S and a generating set C of S, we explicitly determine the color-permutable automorphism group of \(\mathrm {Cay}(S,C)\) [Theorem 1.1]. Also for a monoid S and a generating set C of S, we explicitly determine the color-preserving automorphism group (endomorphism monoid) of \(\mathrm {Cay}(S,C)\) [Proposition 2.3 and Corollary 2.4]. Then we use these results to characterize when \(\mathrm {Cay}(S,C)\) is color-endomorphism vertex-transitive [Theorem 3.4].

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

This paper deals with the Cayley graph Cay(Symn,Tn), where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that Aut(Cay(Symn,Tn)) is the product of the left translation group and a dihedral group Dn+1 of order 2(n+1). The proof uses several properties of the subgraph Γ of Cay(Symn,Tn) induced by the set Tn. In particular, Γ is a 2(n−2)-regular graph whose automorphism group is Dn+1,Γ has as many as n+1 maximal cliques of size 2, and its subgraph Γ(V) whose vertices are those in these cliques is a 3-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of Dn+1 of order n+1 with regular Cayley maps on Symn is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non-t-balanced regular Cayley map on Symn.

Automorphism groups of Cayley graphs generated by connected transposition sets

Let S be a set of transpositions that generates the symmetric group Sn, where n≥3. The transposition graph T(S) is defined to be the graph with vertex set {1,…,n} and with vertices i and j being adjacent in T(S) whenever (i,j)∈S. We prove that if the girth of the transposition graph T(S) is at least 5, then the automorphism group of the Cayley graph Cay(Sn,S) is the semidirect product R(Sn)⋊Aut(Sn,S), where Aut(Sn,S) is the set of automorphisms of Sn that fixes S. This strengthens a result of Feng on transposition graphs that are trees. We also prove that if the transposition graph T(S) is a 4-cycle, then the set of automorphisms of the Cayley graph Cay(S4,S) that fixes a vertex and each of its neighbors is isomorphic to the Klein 4-group and hence is nontrivial. We thus identify the existence of 4-cycles in the transposition graph as being an important factor in causing a potentially larger automorphism group of the Cayley graph.

Automorphism Groups of Rational Circulant Graphs

The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups.

The Cayley graphs of ℤd and the limits of vertex-primitive graphs of HA-type

We study the limits of the finite graphs that admit some vertex-primitive group of automorphisms with a regular abelian normal subgroup. It was shown in [1] that these limits are Cayley graphs of the groups ℤd. In this article we prove that for each d > 1 the set of Cayley graphs of ℤd presenting the limits of finite graphs with vertex-primitive and edge-transitive groups of automorphisms is countable (in fact, we explicitly give countable subsets of these limit graphs). In addition, for d < 4 we list all Cayley graphs of ℤd that are limits of minimal vertex-primitive graphs. The proofs rely on a connection of the automorphism groups of Cayley graphs of ℤd with crystallographic groups.

Automorphism groups of Cayley graphs on symmetric groups with generating transposition sets

Let T be a set of transpositions of the symmetric group S n . The transposition graph Tra ( T ) of T is the graph with vertex set { 1 , 2 , … , n } and edge set { ij | ( i j ) ∈ T } . In this paper it is shown that if n ⩾ 3 , then the automorphism group of the transposition graph Tra ( T ) is isomorphic to Aut ( S n , T ) = { α ∈ Aut ( S n ) | T α = T } and if T is a minimal generating set of S n then the automorphism group of the Cayley graph Cay ( S n , T ) is the semiproduct R ( S n ) ⋊ Aut ( S n , T ) , where R ( S n ) is the right regular representation of S n . As a result, we generalize a theorem of Godsil and Royle [C.D. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001, p. 53] regarding the automorphism groups of Cayley graphs on S n : if T is a minimal generating set of S n and the automorphism group of the transposition graph Tra ( T ) is trivial then the automorphism group of the Cayley graph Cay ( S n , T ) is isomorphic to S n .

On the Automorphism Groups of Cayley Graphs of Finite Simple Groups

Let G be a nite nonabelian simple group and let be a connected undirected Cayley graph for G. The possible structures for the full automorphism group Aut are specied. Then, for certain nite simple groups G, a sucient condition is given under which G is a normal subgroup of Aut. Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups G =J 1 ,J 4, Ly and BM, while others fall into two innite families and involve the Ree simple groups and alternating groups. The two innite families contain examples of half-transitive graphs of arbitrarily large valency.

Automorphisms of Cayley graphs of metacyclic groups of prime-power order

AbstractThis paper inverstigates the automorphism groups of Cayley graphs of metracyclicp-gorups. A characterization is given of the automorphism groups of Cayley grahs of a metacyclicp-group for odd primep. In particular, a complete determiniation of the automophism group of a connected Cayley graph with valency less than 2pof a nonabelian metacyclicp-group is obtained as a consequence. In subsequent work, the result of this paper has been applied to solve several problems in graph theory.

The Structure of Automorphism Groups of Cayley Graphs and Maps

The automorphism groups Aut(C(G, X)) and Aut(CM(G, X, p)) of a Cayley graph C(G, X) and a Cayley map CM(G, X, p) both contain an isomorphic copy of the underlying group G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions of the group G by the stabilizer subgroup of the vertex 1 G . We use this description to derive necessary and sufficient conditions to be satisfied by a finite group in order to be the (full) automorphism group of a Cayley graph or map and classify all the finite groups that can be represented as the (full) automorphism group of some Cayley graph or map.

Automorphisms of Cayley graphs

The aim of this paper is to characterize subgroups of non-regular automorphism groups of Cayley graphs. Relations between group automorphisms, graph automorphisms and permutations of the edge set of Cayley graphs are investigated.

On automorphism groups of Cayley graphs

LetXG,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(XG,H) must contain the regular subgroupLG corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(XG,H)∶LG] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.