Connected Cayley Graph Research Articles

Aldous’ spectral gap property for normal Cayley graphs on symmetric groups

Aldous’ spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on Sn with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of Sn. In this paper, we prove three results on normal Cayley graphs on Sn possessing this property for sufficiently large n, one of which can be viewed as a generalization of the “normal” case of Aldous’ spectral gap conjecture.

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.

On hamiltonian cycles in Cayley graphs of order pqrs

Let G be a finite group. We show that if |G| = pqrs, where p, q, r, and s are distinct odd primes, then every connected Cayley graph on G has a hamiltonian cycle.

Cayley Graphs Defined by Systems of Equations

Let R be a finite ring. In this paper, we mainly explore the conditions to ensure the graph BΓn defined by a system of equations {fi|i=2,…,n} to be a Cayley graph or a Hamiltonian graph. More precisely, we prove that BΓn is a Cayley graph with G=⟨ϕ,A⟩ a group of dihedral type if and only if the system Fn={fi|i=2,…,n} is Cayley graphic of dihedral type in R. As an application, the well-known Lova´sz Conjecture, which states that any finite connected Cayley graph has a Hamilton cycle, holds for the connected BΓn defined by Cayley graphic system Fn of dihedral type in the field GF(pk).

A finite simple group is CCA if and only if it has no element of order four

A Cayley graph for a group G is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of G is an element of the normaliser of G. A group G is then said to be CCA if every connected Cayley graph on G is CCA. We show that a finite simple group is CCA if and only if it has no element of order 4. We also show that “many” 2-groups are non-CCA.

Cayley graphs of order kp are hamiltonian for k < 48

We provide a computer-assisted proof that if G is any finite group of order kp, where k < 48 and p is prime, then every connected Cayley graph on G is hamiltonian (unless kp = 2). As part of the proof, it is verified that every connected Cayley graph of order less than 48 is either hamiltonian connected or hamiltonian laceable (or has valence less than three).

Hamiltonian Cycles in Normal Cayley Graphs

It has been conjecture that every finite connected Cayley graph contains a hamiltonian cycle. Given a finite group G and a connection set S, the Cayley graph Cay(G, S) will be called normal if for every $$g\in G$$ we have that $$g^{-1}Sg = S$$. In this paper we present some conditions on the order of the elements of the connexion set which imply the existence of a hamiltonian cycle in the graph and we construct it in an explicit way.

An improved diameter bound for finite simple groups of Lie type

For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a non-abelian finite simple group. Let $G$ be a finite simple group of Lie type of Lie rank $n$ over the field $F_{q}$. Babai's conjecture has been verified in case $n$ is bounded, but it is wide open in case $n$ is unbounded. Recently, Biswas and Yang proved that $\mathrm{diam}(G)$ is bounded by $q^{O( n {(\log_{2}n + \log_{2}q)}^{3})}$. We show that in fact $\mathrm{diam}(G) < q^{O(n {(\log_{2}n)}^{2})}$ holds. Note that our bound is significantly smaller than the order of $G$ for $n$ large, even if $q$ is large. As an application, we show that more generally $\mathrm{diam}(H) < q^{O( n {(\log_{2}n)}^{2})}$ holds for any subgroup $H$ of $\mathrm{GL}(V)$, where $V$ is a vector space of dimension $n$ defined over the field $F_q$.

Normal edge-transitive and \frac{1}{2}-arc-transitive cayley graphs on non-abelian groups of odd order 3pq, p and q are primes

Suppose $p$ and $q$ are odd prime numbers. In this paper, the connected Cayley graph of groups of order $3pq$, for primes $p$ and $q$, are investigated and all connected normal $\frac{1}{2}-$arc-transitive Cayley graphs of group of these orders will be classified.

Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian

We show that if G is a finite group whose commutator subgroup [G, G] has order 2p, where p is an odd prime, then every connected Cayley graph on G has a hamiltonian cycle.

Characterising CCA Sylow cyclic groups whose order is not divisible by four

A Cayley graph on a group G has a natural edge-colouring. We say that such a graph is CCA if every automorphism of the graph that preserves this edge-colouring is an element of the normaliser of the regular representation of G . A group G is then said to be CCA if every connected Cayley graph on G is CCA. Our main result is a characterisation of non-CCA graphs on groups that are Sylow cyclic and whose order is not divisible by four. We also provide several new constructions of non-CCA graphs.

Infinitely many nonsolvable groups whose Cayley graphs are hamiltonian

We show there are infinitely many finite groups~$G$, such that every connected Cayley graph on~$G$ has a hamiltonian cycle, and $G$ is not solvable. Specifically, we show that if $A_5$~is the alternating group on five letters, and $p$~is any prime, such that $p \equiv 1 \pmod{30}$, then every connected Cayley graph on the direct product $A_5 \times \integer _p$ has a hamiltonian cycle.

On colour-preserving automorphisms of Cayley graphs

We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G , such that every such automorphism of every connected Cayley graph on G has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.

Hamilton decompositions of 6-regular Cayley graphs on even Abelian groups with involution-free connections sets

Alspach conjectured that every connected Cayley graph on a finite Abelian group A is Hamilton-decomposable. Liu has shown that for |A| even, if S={s1,…,sk}⊂A is an inverse-free strongly minimal generating set of A, then the Cayley graph Cay(A;S⋆), is decomposable into k Hamilton cycles, where S⋆ denotes the inverse-closure of S. Extending these techniques and restricting to the 6-regular case, this article relaxes the constraint of strong minimality on S to require only that S be strongly a-minimal, for some a∈S and the index of 〈a〉 be at least four. Strong a-minimality means that 2s∉〈a〉 for all s∈S∖{a,−a}. Some infinite families of open cases for the 6-regular Cayley graphs on even order Abelian groups are resolved. In particular, if |s1|≥|s2|>2|s3|, then Cay(A;{s1,s2,s3}⋆) is Hamilton-decomposable.

Odd-order Cayley graphs with commutator subgroup of order pq are hamiltonian

We show that if G is a nontrivial, finite group of odd order, whose commutator subgroup [ G , G ] is cyclic of order p μ q ν , where p and q are prime, then every connected Cayley graph on G has a hamiltonian cycle.

Rotational circulant graphs

A Frobenius group is a transitive permutation group which is not regular but only the identity element can fix two points. Such a group can be expressed as the semidirect product G=K⋊H of a nilpotent normal subgroup K and another group H fixing a point. A first-kind G-Frobenius graph is a connected Cayley graph on K with connection set an H-orbit aH on K that generates K, where H has an even order or a is an involution. It is known that the first-kind Frobenius graphs admit attractive routing and gossiping algorithms. A complete rotation in a Cayley graph on a group G with connection set S is an automorphism of G fixing S setwise and permuting the elements of S cyclically. It is known that if the fixed-point set of such a complete rotation is an independent set and not a vertex-cut, then the gossiping time of the Cayley graph (under a certain model) attains the smallest possible value. In this paper we classify all first-kind Frobenius circulant graphs that admit complete rotations, and describe a means to construct them. This result can be stated as a necessary and sufficient condition for a first-kind Frobenius circulant to be 2-cell embeddable on a closed orientable surface as a balanced regular Cayley map. We construct a family of non-Frobenius circulants admitting complete rotations such that the corresponding fixed-point sets are independent and not vertex-cuts. We also give an infinite family of counterexamples to the conjecture that the fixed-point set of every complete rotation of a Cayley graph is not a vertex-cut.

Formula omitted]-connectivity in Abelian Cayley graphs

Let G be a 2-edge-connected undirected graph, A be an (additive) Abelian group and A∗=A−{0}. A graph G is A-connected if G has an orientation G′ such that for every map b:V(G)↦A satisfying ∑v∈V(G)b(v)=0, there is a function f:E(G)↦A∗ such that for each vertex v∈V(G), the total amount of f-values on the edges directed out from v minus the total amount of f-values on the edges directed into v equals b(v). Jaeger et al. [F. Jaeger, N. Linial, C. Payan, M. Tarsi, Group connectivity of graphs—a nonhomogeneous analogue of nowhere-zero flow properties, J. Combinatorial Theory, Series B 56 (1992) 165–182] conjectured that every 5-edge-connected graph G is Z3-connected, where Z3 is the cyclic group of order 3. In this paper we prove that every connected Cayley graph G of degree at least 5 on an Abelian group is Z3-connected.

Cayley graphs on nilpotent groups with cyclic commutator subgroup are hamiltonian

We show that if G is any nilpotent, finite group, and the commutator subgroup of G is cyclic, then every connected Cayley graph on G has a hamiltonian cycle.

On Cubic Nonsymmetric Cayley Graphs

Let be a connected Cayley graph of group G, then Γ is called normal if the right regular representation of G is a normal subgroup of , the full automorphism group of Γ. For the case where G is a finite nonabelian simple group and Γ is symmetric cubic Cayley graph, Caiheng Li and Shangjin Xu proved that Γ is normal with only two exceptions. Since then, the normality of nonsymmetric cubic Cayley graph of nonabelian simple group aroused strong interest of people. So far such graphs which have been known are all normal. Then people conjecture that all of such graphs are either normal or the Cayley subset consists of involutions. In this paper we give an negative answer by two counterexamples. As far as we know these are the first examples for the non-normal cubic nonsymmetric Cayley graphs of finite nonabelian simple groups.

On 4-valent Frobenius circulant graphs

Graph Theory A 4-valent first-kind Frobenius circulant graph is a connected Cayley graph DLn(1, h) = Cay(Zn, H) on the additive group of integers modulo n, where each prime factor of n is congruent to 1 modulo 4 and H = {[1], [h], −[1], −[h]} with h a solution to the congruence equation x 2 + 1 ≡ 0 (mod n). In [A. Thomson and S. Zhou, Frobenius circulant graphs of valency four, J. Austral. Math. Soc. 85 (2008), 269-282] it was proved that such graphs admit 'perfect ' routing and gossiping schemes in some sense, making them attractive candidates for modelling interconnection networks. In the present paper we prove that DLn(1, h) has the smallest possible broadcasting time, namely its diameter plus two, and we explicitly give an optimal broadcasting in DLn(1, h). Using number theory we prove that it is possible to recursively construct larger 4-valent first-kind Frobenius circulants from smaller ones, and we give a methodology for such a construction. These and existing results suggest that, among all 4-valent circulant graphs, 4-valent first-kind Frobenius circulants are extremely efficient in terms of routing, gossiping, broadcasting and recursive construction.