Edge-transitive Cayley Graphs Research Articles

On sensitivity in bipartite Cayley graphs

Huang proved that every set of more than half the vertices of the d-dimensional hypercube Qd induces a subgraph of maximum degree at least d, which is tight by a result of Chung, Füredi, Graham, and Seymour. Huang asked whether similar results can be obtained for other highly symmetric graphs.First, we present three infinite families of Cayley graphs of unbounded degree that contain induced subgraphs of maximum degree 1 on more than half the vertices. In particular, this refutes a conjecture of Potechin and Tsang, for which first counterexamples were shown recently by Lehner and Verret. The first family consists of dihedrants and contains a sporadic counterexample encountered earlier by Lehner and Verret. The second family are star graphs, these are edge-transitive Cayley graphs of the symmetric group. All members of the third family are d-regular containing an induced matching on a d2d−1-fraction of the vertices. This is largest possible and answers a question of Lehner and Verret.Second, we consider Huang's lower bound for graphs with subcubes and show that the corresponding lower bound is tight for products of Coxeter groups of type An, I2(2k+1), and most exceptional cases. We believe that Coxeter groups are a suitable generalization of the hypercube with respect to Huang's question.Finally, we show that induced subgraphs on more than half the vertices of Levi graphs of projective planes and of the Ramanujan graphs of Lubotzky, Phillips, and Sarnak have unbounded degree. This gives classes of Cayley graphs with properties similar to the ones provided by Huang's results. However, in contrast to Coxeter groups these graphs have no subcubes.

Normal edge-transitive Cayley graphs and Frattini-like subgroups

For a finite group G and an inverse-closed generating set C of G, let Aut(G;C) consist of those automorphisms of G which leave C invariant. We define an Aut(G;C)-invariant normal subgroup Φ(G;C) of G which has the property that, for any Aut(G;C)-invariant normal set of generators for G, if we remove from it all the elements of Φ(G;C), then the remaining set is still an Aut(G;C)-invariant normal generating set for G. The subgroup Φ(G;C) contains the Frattini subgroup Φ(G) but the inclusion may be proper. The Cayley graph Cay(G,C) is normal edge-transitive if Aut(G;C) acts transitively on the pairs {c,c−1} from C. We show that, for a normal edge-transitive Cayley graph Cay(G,C), its quotient modulo Φ(G;C) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ(G;C) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi.

Tetravalent edge-transitive Cayley graphs of Frobenius groups

In this paper, we give a characterisation for a class of edge-transitive Cayley graphs and provide a method for constructing edge-transitive graphs of valency 4 with arbitrarily large vertex stabiliser. In particular, in the last section, we obtain certain extensions of the results of Li et al. (Tetravalent edge-transitive Cayley graphs with odd number of vertices, J Comb Theory Ser B 96:164–181, 2006) on half-transitive graphs.

On Reconstruction of Normal Edge-Transitive Cayley Graphs

The main idea of this paper is to provide an algebraic algorithm for constructing symmetric graphs with optimal fault tolerance. For this purpose, we use normal edge-transitive Cayley graphs and the idea of reconstruction question posed by Praeger to present a special factorization of groups which induces a graphical decomposition of normal edge-transitive Cayley graphs to simpler normal edge-transitive Cayley graphs. Then as a consequence of our results, we continue the study of normal edge-transitive Cayley graphs of abelian groups and we show that knowing normal edge-transitive Cayley graphs of abelian p-groups, we can determine all normal edge-transitive Cayley graphs of abelian groups.

A family of edge-transitive Cayley graphs

Edge-transitive graphs of order a prime or a product of two distinct primes with any positive integer valency, and of square-free order with valency at most 7 have been classified by a series of papers. In this paper, a complete classification is given of edge-transitive Cayley graphs of square-free order with valency less than the smallest prime divisor of the order. This leads to new constructions of infinite families of both arc-regular Cayley graphs and edge-regular Cayley graphs (so half-transitive). Also, as by-products, it is proved that, for any given positive integers $$k,s\ge 1$$ and $$m,n\ge 2$$ , there are infinitely many arc-regular normal circulants of valency 2k and order a product of s primes, and there are infinitely many edge-regular normal metacirculants of valency 2m and order a product of n primes; such arc-regular and edge-regular examples are also specifically constructed.

RELATIVE ELEMENTARY ABELIAN GROUPS AND A CLASS OF EDGE-TRANSITIVE CAYLEY GRAPHS

Motivated by a problem of characterising a family of Cayley graphs, we study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\mathsf{Aut}(G)$. It is shown that such groups correspond to complete multipartite graphs which are normal edge-transitive Cayley graphs.

Normal edge-transitive Cayley graphs of Frobenius groups

A Cayley graph for a group G is called normal edge-transitive if it admits an edge-transitive action of some subgroup of the holomorph of G [the normaliser of a regular copy of G in $${{\mathrm{Sym}}}(G)$$Sym(G)]. We complete the classification of normal edge-transitive Cayley graphs of order a product of two primes by dealing with Cayley graphs for Frobenius groups of such orders. We determine the automorphism groups of these graphs, proving in particular that there is a unique vertex-primitive example, namely the flag graph of the Fano plane.

Normal Edge-Transitive Cayley Graphs of the Group U6n

A Cayley graph of a group G is called normal edge-transitive if the normalizer of the right representation of the group in the automorphism of the Cayley graph acts transitively on the set of edges of the graph. In this paper, we determine all connected normal edge-transitive Cayley graphs of the group U6n.

A characterization of a family of edge-transitive metacirculant graphs

A characterization is given of the class of edge-transitive Cayley graphs of Frobenius groups Zrd:Zm with r an odd prime and m odd, of valency less than 2p1 with p1 the smallest prime divisor of m. It is shown that either (rd,m)=(p,p−12)or(29,7), or such a graph is a normal Cayley graph and half-transitive. This provides new construction of half-transitive graphs.

Tetravalent edge-transitive cayley graphs of PGL (2, p)

Let Γ = Cay(G, S), R(G) be the right regular representation of G. The graph Γ is called normal with respect to G, if R(G) is normal in the full automorphism group Aut(Γ) of Γ. Γ is called a bi-normal with respect to G if R(G) is not normal in Aut(Γ), but R(G) contains a subgroup of index 2 which is normal in Aut(Γ). In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL(2, p) are either normal or bi-normal when p ≠ 11 is a prime.

Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number

We determine all connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p, where p is a prime number. As a consequence we prove if |G| = 2δp, δ = 0, 1,2 and p prime, then Λ = Cay(G, S) is a connected normal \(\tfrac{1} {2}\) arc-transitive Cayley graph only if G = F4p, where S is an inverse closed generating subset of G which does not contain the identity element of G and F4p is a group with presentation \(F_{4p} = \left\langle {\left. {a,b} \right|a^p = b^4 = 1,b^{ - 1} ab = a^\lambda } \right\rangle\), where λ2 ≠ −1 (mod p).

Classifying a family of edge-transitive metacirculant graphs

A characterization is given of the class of edge-transitive Cayley graphs of Frobenius groups $\mathbb{Z}_{p^{d}}{:}\mathbb{Z}_{q}$ with p,q odd prime, of valency coprime to p. This characterization is then used to study an isomorphism problem regarding Cayley graphs, and to construct new families of half-arc-transitive graphs.

Nonnormal Edge-Transitive Cubic Cayley Graphs of Dihedral Groups

A Cayley graph of a finite group is called normal edge transitive if its automorphism group has a subgroup which both normalizes and acts transitively on edges. In this paper we determine all cubic, connected, and undirected edge-transitive Cayley graphs of dihedral groups, which are not normal edge transitive. This is a partial answer to the question of Praeger (1999).

Some Normal Edge-transitive Cayley Graphs On Dihedral Groups

Let \(G\) be a group and \(S\) a subset of \(G\) such that \(1_G\not\in S\) and \(S=S−1\). Let \(\Gamma=Cay(G,S)\) be a Cayley graph on \(G\) relative to. Then \(\Gamma\) is said to be normal edge-transitive, if \(N_{Aut}(\Gamma)(G)\) acts transitively on edges. In this paper we determine all normal edge-transitive Cayley graphs on a dihedral Group \(D_{2n}\) of valency \(n\). In addition we classify normal edge-transitive Cayley graphs \(\Gamma=Cay(D_{2p},S)\) of valency four, for a prime \(p\) and give some normal edge-transitive Cayley graphs \(\Gamma=Cay(D_{2n},S)\) of valency four that \(n\) is not a prime .

On automorphism groups of quasiprimitive 2-arc transitive graphs

We characterize the automorphism groups of quasiprimitive 2-arc-transitive graphs of twisted wreath product type. This is a partial solution for a problem of Praeger regarding quasiprimitive 2-arc transitive graphs. The solution stimulates several further research problems regarding automorphism groups of edge-transitive Cayley graphs and digraphs.

Finite edge-transitive Cayley graphs and rotary Cayley maps

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups. In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

Tetravalent edge-transitive Cayley graphs with odd number of vertices

A characterisation is given of edge-transitive Cayley graphs of valency 4 on odd number of vertices. The characterisation is then applied to solve several problems in the area of edge-transitive graphs: answering a question proposed by Xu [Automorphism groups and isomorphisms of Cayley graphs, Discrete Math. 182 (1998) 309–319] regarding normal Cayley graphs; providing a method for constructing edge-transitive graphs of valency 4 with arbitrarily large vertex-stabiliser; constructing and characterising a new family of half-transitive graphs. Also this study leads to a construction of the first family of arc-transitive graphs of valency 4 which are non-Cayley graphs and have a ‘nice’ isomorphic 2-factorisation.

On edge-transitive Cayley graphs of valency four

A characterization is given of a class of edge-transitive Cayley graphs, providing methods for constructing Cayley graphs with certain symmetry properties. Various new half-arc transitive graphs are constructed.

Finite normal edge-transitive Cayley graphs

An approach to analysing the family of Cayley graphs for a finite group G is given which identifies normal edge-transitive Cayley graphs as a sub-family of central importance. These are the Cayley graphs for G for which a subgroup of automorphisms exists which both normalises G and acts transitively on edges. It is shown that, for a nontrivial group G, each normal edge-transitive Cayley graph for G has at least one homomorphic image which is a normal edge-transitive Cayley graph for a characteristically simple quotient group of G. Moreover, given a normal edge-transitive Cayley graph ΓH for a quotient group G/H, necessary and sufficient conditions are obtained for the existence of a normal edge-transitive Cayley graph Γ for G which has ΓH as a homomorphic image, and a method for obtaining all such graphs Γ is given.