Transitive Cayley Research Articles

On transitive Cayley graphs of homogeneous inverse semigroups

On transitive Cayley graphs of homogeneous inverse semigroups

On Finite Subnormal Cayley Graphs

In this paper we introduce and study a type of Cayley graph – subnormal Cayley graph. We prove that a subnormal 2-arc transitive Cayley graph is a normal Cayley graph or a normal cover of a complete bipartite graph $\mathbf{K}_{p^d,p^d}$ with $p$ prime. Then we obtain a generic method for constructing half-symmetric (namely edge transitive but not arc transitive) Cayley graphs.

On transitive Cayley graphs of pseudo-unitary homogeneous semigroups

We characterize colour-preserving automorphism vertex transitivity and vertex transitivity of the Cayley graphs of all semigroups in a class of pseudo-unitary homogeneous semigroups.

CLASSIFICATION OF TETRAVALENT -TRANSITIVE NONNORMAL CAYLEY GRAPHS OF FINITE SIMPLE GROUPS

AbstractA graph $\Gamma $ is called $(G, s)$ -arc-transitive if $G \le \text{Aut} (\Gamma )$ is transitive on the set of vertices of $\Gamma $ and the set of s-arcs of $\Gamma $ , where for an integer $s \ge 1$ an s-arc of $\Gamma $ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots ,v_s)$ of $\Gamma $ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$ . A graph $\Gamma $ is called 2-transitive if it is $(\text{Aut} (\Gamma ), 2)$ -arc-transitive but not $(\text{Aut} (\Gamma ), 3)$ -arc-transitive. A Cayley graph $\Gamma $ of a group G is called normal if G is normal in $\text{Aut} (\Gamma )$ and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved that if $\Gamma $ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either $\Gamma $ is normal or G is one of the groups $\text{PSL}_2(11)$ , ${\rm M} _{11}$ , $\text{M} _{23}$ and $A_{11}$ . However, it was unknown whether $\Gamma $ is normal when G is one of these four groups. We answer this question by proving that among these four groups only $\text{M} _{11}$ produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

ON CONNECTED TETRAVALENT NORMAL EDGE-TRANSITIVE CAYLEY GRAPHS OF NON-ABELIAN GROUPS OF ORDER 5p^2

Our aim in this paper is to investigate graph automorphism and group automorphism determining all connected tetravalent normal edge transitive Cayley graphs on non-Abelian groups of order 5p2 with respect to tetravalent sets and same-order elements, where p is a prime number and its Sylow p-subgroup is cyclic.

On Classification of 2-Arc Transitive Cayley Graphs of the Dicyclic Group

In this paper we first determine all possible connected core-free 2-arc transitive Cayley graphs of the dicyclic group, $$B_{4n}$$ , and then show that this can be used to classify all connected 2-arc transitive Cayley graphs of this group in terms of regular cyclic covers, provided that we also know connected core-free 2-arc transitive Cayley graphs of the dihedral group.

FINITE NORMAL 2-GEODESIC TRANSITIVE CAYLEY GRAPHS

For an odd prime $p$, a $p$-transposition group is a group generated by a set of involutions such that the product of any two has order 2 or $p$. We first classify a family of $(G,2)$-geodesic transitive Cayley graphs ${\rm\Gamma}:=\text{Cay}(T,S)$ where $S$ is a set of involutions and $T:\text{Inn}(T)\leq G\leq T:\text{Aut}(T,S)$. In this case, $T$ is either an elementary abelian 2-group or a $p$-transposition group. Then under the further assumption that $G$ acts quasiprimitively on the vertex set of ${\rm\Gamma}$, we prove that: (1) if ${\rm\Gamma}$ is not $(G,2)$-arc transitive, then this quasiprimitive action is the holomorph affine type; (2) if $T$ is a $p$-transposition group and $S$ is a conjugacy class, then $p=3$ and ${\rm\Gamma}$ is $(G,2)$-arc transitive.

Finite 2-Geodesic Transitive Abelian Cayley Graphs

In this paper, we first give a classification of the family of 2-geodesic transitive abelian Cayley graphs. Let $$\Gamma $$Γ be such a graph which is not 2-arc transitive. It is shown that one of the following holds: (1) $$\Gamma \cong \mathrm{K}_{m[b]}$$Γ?Km[b] for some $$m\ge 3$$m?3 and $$b\ge 2$$b?2; (2) $$\Gamma $$Γ is a normal Cayley graph of an elementary abelian group; (3) $$\Gamma $$Γ is a cover of Cayley graph $$\Gamma _K$$ΓK of an abelian group T / K, where either $$\Gamma _K$$ΓK is complete arc transitive or $$\Gamma _K$$ΓK is 2-geodesic transitive of girth 3, and A / K acts primitively on $$V(\Gamma _K)$$V(ΓK) of type Affine or Product Action. Second, we completely determine the family of 2-geodesic transitive circulants.

On color-automorphism vertex transitivity of semigroups

In this paper, we continue the study of Kelarev and Praeger devoted to the color-automorphism vertex transitivity of Cayley graphs of semigroups and we generalize and complete some of their results. For this purpose, first we show that for a semigroup S and a non-empty subset C⊆S, the ColAutC(S)-vertex-transitivity of Cay(S,C) is equivalent to the ColAut〈C〉(S)-vertex transitivity of Cay(S,〈C〉), where 〈C〉 denotes the subsemigroup generated by C in S. Then we use this result to characterize a color-automorphism vertex transitive Cayley graph Cay(S,C), where for every a∈S, 〈C〉a is a simple 〈C〉-act or for every a∈S, 〈C〉a is finite. Similarly, we characterize a ColAutC(S)-vertex-transitive Cay(S,C) when for every c∈C, |〈c〉| is infinite and c is left cancellable. Finally, we use these results to establish that if S=∪̇α∈YSα is a semilattice of semigroups Sα and C is a non-empty subset of S, then the ColAutC(S)-vertex-transitivity of Cay(S,C) implies that Y has an identity e and C=Ce. This answers an open question asked in a previous article.

On normal 2-geodesic transitive Cayley graphs

We investigate connected normal 2-geodesic transitive Cayley graphs Cay(T,S). We first prove that if Cay(T,S) is neither cyclic nor K4[2], then ?a??{1}⊆?S for all a?S. Next, as an application, we give a reduction theorem proving that each graph in this family which is neither a complete multipartite graph nor a bipartite 2-arc transitive graph, has a normal quotient that is either a complete graph or a Cayley graph in the family for a characteristically simple group. Finally we classify complete multipartite graphs in the family.

Generalization of Transitive Cayley Digraphs

This paper proves several extended theoretical results of transitive Cayley digraphs. Several generalization of transitive Cayley digraphs also have been provided. Moreover, various graph properties have been expressed in terms of algebraic properties. This did not attract much attention in the literature.

Cubic [formula omitted]-arc transitive Cayley graphs

This paper gives a characterization of connected cubic s -transitive Cayley graphs. It is shown that, for s ≥ 3 , every connected cubic s -transitive Cayley graph is a normal cover of one of 13 graphs: three 3-transitive graphs, four 4-transitive graphs and six 5-transitive graphs. Moreover, the argument in this paper also gives another proof for a well-known result which says that all connected cubic arc-transitive Cayley graphs of finite non-abelian simple groups are normal except two 5-transitive Cayley graphs of the alternating group A 47 .

On transitive Cayley graphs of strong semilattices of right (left) groups

We investigate Cayley graphs of strong semilattices of right (left) groups, of right (left) zero semigroups, and of groups. We show under which conditions they enjoy the property of automorphism vertex transitivity in analogy to Cayley graphs of groups.

On finite edge transitive graphs and rotary maps

It is shown that every connected vertex and edge transitive graph has a normal multicover that is a connected normal edge transitive Cayley graph. Moreover, every chiral or regular map has a normal cover that is a balanced chiral or regular Cayley map, respectively. As an application, a new family of half-transitive graphs is constructed as 2-fold covers of a family of 2-arc transitive graphs admitting Suzuki groups.

Finite 𝑠-arc transitive Cayley graphs and flag-transitive projective planes

In this paper, a characterisation is given of finite s s -arc transitive Cayley graphs with s ≥ 2 s\ge 2 . In particular, it is shown that, for any given integer k k with k ≥ 3 k\ge 3 and k ≠ 7 k\not =7 , there exists a finite set (maybe empty) of s s -transitive Cayley graphs with s ∈ { 3 , 4 , 5 , 7 } s\in \{3,4,5,7\} such that all s s -transitive Cayley graphs of valency k k are their normal covers. This indicates that s s -arc transitive Cayley graphs with s ≥ 3 s\ge 3 are very rare. However, it is proved that there exist 4-arc transitive Cayley graphs for each admissible valency (a prime power plus one). It is then shown that the existence of a flag-transitive non-Desarguesian projective plane is equivalent to the existence of a very special arc transitive normal Cayley graph of a dihedral group.

On cubic s-arc transitive Cayley graphs of finite simple groups

For a positive integer s, a graph Γ is called s- arc transitive if its full automorphism group Aut Γ acts transitively on the set of s-arcs of Γ. Given a group G and a subset S of G with S= S −1 and 1∉ S, let Γ=Cay( G, S) be the Cayley graph of G with respect to S and G R the set of right translations of G on G. Then G R forms a regular subgroup of Aut Γ. A Cayley graph Γ=Cay( G, S) is called normal if G R is normal in Aut Γ. In this paper we investigate connected cubic s-arc transitive Cayley graphs Γ of finite non-Abelian simple groups. Based on Li’s work (Ph.D. Thesis (1996)), we prove that either Γ is normal with s≤2 or G= A 47 with s=5 and Aut Γ ≅ A 48. Further, a connected 5-arc transitive cubic Cayley graph of A 47 is constructed.

On transitive Cayley graphs of groups and semigroups

We investigate Cayley graphs of semigroups and show that they sometimes enjoy properties analogous to those of the Cayley graphs of groups.