Normal Cayley Graphs Research Articles

Improved covering results for conjugacy classes of symmetric groups via hypercontractivity

Abstract We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as normal sets. We show that for any $\epsilon>0$ , there exists $n_0$ such that if $n>n_0$ and A is a normal subset of the symmetric group $S_n$ of density $\ge e^{-n^{2/5 - \epsilon }}$ , then $A^2 \supseteq A_n$ . This improves upon a seminal result of Larsen and Shalev (Inventiones Math., 2008), with our $2/5$ in the double exponent replacing their $1/4$ . Our proof strategy combines two types of techniques. The first is ‘traditional’ techniques rooted in character bounds and asymptotics for the Witten zeta function, drawing from the foundational works of Liebeck–Shalev, Larsen–Shalev, and more recently, Larsen–Tiep. The second is a sharp hypercontractivity theorem in the symmetric group, which was recently obtained by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies not only allows us to attain our improved bounds but also provides new insights into the behavior of general independent sets in normal Cayley graphs over symmetric groups.

A note on eigenvalues of Cayley graphs

A graph is called integral if all its eigenvalues are integers. A Cayley graph is called normal if its connection set is a union of conjugacy classes. We show that a non-empty integral normal Cayley graph for a group of odd order has an odd eigenvalue.

Two-geodesic transitive graphs of order pn with n ≤ 3

A vertex triple (u,v,w) of a graph is called a 2-geodesic if v is adjacent to both u and w and u is not adjacent to w. A graph is said to be 2-geodesic transitive if its automorphism group is transitive on the set of 2-geodesics. In this paper, a complete classification of 2-geodesic transitive graphs of order pn is given for each prime p and n≤3. It turns out that all such graphs consist of three small graphs: the complete bipartite graph K4,4 of order 8, the Schläfli graph of order 27 and its complement, and fourteen infinite families: the cycles Cp,Cp2 and Cp3, the complete graphs Kp,Kp2 and Kp3, the complete multipartite graphs Kp[p], Kp[p2] and Kp2[p], the Hamming graph H(2,p) and its complement, the Hamming graph H(3,p), and two infinite families of normal Cayley graphs on the extraspecial group of order p3 and exponent p.

Normal Cayley digraphs of dihedral groups with CI-property

A Cayley (di)graph Cay(G,S) of a group G with respect to a set S ⊆ G is said to be normal if the image of G under its right regular representation is normal in the automorphism group of Cay(G,S), and is called a CI-(di)graph if for every T ⊆ G with Cay(G,S)≅ Cay(G,T), there is α∈ Aut(G) such that Sα = T. A finite group G is called a DCI-group or an NDCI-group if all Cayley digraphs or all normal Cayley digraphs of G are CI-digraphs, respectively, and is called a CI-group or an NCI-group if all Cayley graphs or all normal Cayley graphs of G are CI-graphs, respectively. Motivated by a conjecture proposed by Ádám in 1967, CI-groups and DCI-groups have been actively studied during the last fifty years by many researchers in algebraic graph theory. It took about thirty years to obtain the classification of cyclic CI-groups and DCI-groups, and recently, the first two authors, among others, classified cyclic NCI-groups and NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCI-groups, their classification is still elusive. In this paper, we prove that a dihedral group of order 2n is an NCI-group or an NDCI-group if and only if n = 2, 4 or n is odd. As a direct consequence, we have that if a dihedral group D2n of order 2n is a DCI-group then n = 2 or n is odd-square-free, and that if D2n is a CI-group then n = 2, 9 or n is odd-square-free, throwing some new light on classification of dihedral CI-groups and DCI-groups. As a byproduct, we construct a non-CI Cayley graph of the dihedral group D8, but Holt and Royle [A census of small transitive groups and vertex-transitive graphs, J. Symbolic Comput. 101 (2020) 51-60] claimed that D8 is a CI-group by an algorithm there.

Strongly cospectral vertices in normal Cayley graphs

We prove an upper bound on the number of pairwise strongly cospectral vertices in a normal Cayley graph, in terms of the multiplicities of its eigenvalues. We use this to determine an explicit bound in Cayley graphs of Z2d and Z4d. We also provide some infinite families of Cayley graphs of Z2d with a set of four pairwise strongly cospectral vertices and show that such graphs exist in every dimension.

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.

Continuous-time Quantum Walks on Cayley Graphs of Extraspecial Groups

We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state transfer and fractional revival. Using partial spreads, we construct Cayley graphs on extraspecial 2-groups that admit these phenomena. We also show that there is no normal Cayley graph of an extraspecial group that admits instantaneous uniform mixing.

Two-distance transitive normal Cayley graphs

In this paper, we construct an infinite family of normal Cayley graphs, which are $2$-distance-transitive but neither distance-transitive nor $2$-arc-transitive. This answers a question raised by Chen, Jin and Li in 2019 and corrects a claim in a literature given by Pan, Huang and Liu in 2015.

Normal Cayley digraphs of cyclic groups with CI-property

A Cayley (di)graph of a group G is called normal if the right regular representation of G is normal in the full automorphism group of and a CI-(di)graph if for every Cayley (di)graph implies that there is such that We call a group G an NDCI-group or NCI-group if all normal Cayley digraphs or graphs of G are CI-digraphs or CI-graphs, respectively. We prove that a cyclic group of order n is an NDCI-group if and only if and an NCI-group if and only if either n = 8 or

Arc-Transitive Graphs of Square-free Order with Valency 11

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

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.

Frobenius REA-groups and their Cayley graphs

In this paper, we determine the automorphism groups of a class of Frobenius groups, and then solve that under what condition they are REA-groups. As an application, we construct a type of normal edge-transitive Cayley graph.

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.

The Second Eigenvalue of some Normal Cayley Graphs of Highly Transitive Groups

Let $G$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $\Gamma=\mathrm{Cay}(G,T)$ be a Cayley graph of $G$. The graph $\Gamma$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $\Gamma$ in terms of the second eigenvalues of certain subgraphs of $\Gamma$. Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $S_n$, and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $S_n$ with $\max_{\tau\in T}|\mathrm{supp}(\tau)|\leqslant 5$, where $\mathrm{supp}(\tau)$ is the set of points in $[n]$ non-fixed by $\tau$.

Constructing 2-Arc-Transitive Covers of Hypercubes

We introduce the notion of a symmetric basis of a vector space equipped with a quadratic form, and provide a sufficient and necessary condition for the existence to such a basis. Symmetric bases are then used to study Cayley graphs of certain extraspecial 2-groups of order 2^{2r+1} (r\geq 1), which are further shown to be normal Cayley graphs and 2-arc-transitive covers of 2r-dimensional hypercubes.

Quasi-semiregular automorphisms of cubic and tetravalent arc-transitive graphs

A non-trivial automorphism g of a graph Γ is called semiregular if the only power gi fixing a vertex is the identity mapping, and it is called quasi-semiregular if it fixes one vertex and the only power gi fixing another vertex is the identity mapping. In this paper, we prove that K4, the Petersen graph and the Coxeter graph are the only connected cubic arc-transitive graphs admitting a quasi-semiregular automorphism, and K5 is the only connected tetravalent 2-arc-transitive graph admitting a quasi-semiregular automorphism. It will also be shown that every connected tetravalent G-arc-transitive graph, where G is a solvable group containing a quasi-semiregular automorphism, is a normal Cayley graph of an abelian group of odd order.

Arc-transitive Cayley graphs on non-abelian simple groups with soluble vertex stabilizers and valency seven

In this paper, we study arc-transitive Cayley graphs on non-abelian simple groups with soluble stabilizers and valency seven. Let Γ be such a Cayley graph on a non-abelian simple group T. It is proved that either Γ is a normal Cayley graph or Γ is S-arc-transitive, with (S,T)=(An,An−1) and n=7,21,63 or 84; and, for each of these four values of n, there really exist some arc-transitive 7-valent non-normal Cayley graphs on An−1 and specific examples are constructed.

Cubic core-free symmetric m-Cayley graphs

An m-Cayley graph $$\Gamma $$ over a group G is defined as a graph which admits G as a semi-regular group of automorphisms with m orbits. This generalises the notions of a Cayley graph (where $$m = 1$$ ) and a bi-Cayley graph (where $$m = 2$$ ). The m-Cayley graph $$\Gamma $$ over G is said to be normal if G is normal in the automorphism group $$\mathrm{Aut}(\Gamma )$$ of $$\Gamma $$ , and core-free if the largest normal subgroup of $$\mathrm{Aut}(\Gamma )$$ contained in G is the identity subgroup. In this paper, we investigate properties of symmetric m-Cayley graphs in the special case of valency 3, and use these properties to develop a computational method for classifying connected cubic core-free symmetric m-Cayley graphs. We also prove that there is no 3-arc-transitive normal Cayley graph or bi-Cayley graph (with valency 3 or more), which answers a question posed by Li (Proc Amer Math Soc 133:31–41 2005). Using our classification method, we give a new proof of the fact that there are exactly 15 connected cubic core-free symmetric Cayley graphs, two of which are Cayley graphs over non-abelian simple groups. We also show that there are exactly 109 connected cubic core-free symmetric bi-Cayley graphs, 48 of which are bi-Cayley graphs over non-abelian simple groups, and that there are 1, 6, 81, 462 and 3267 connected cubic core-free 1-arc-regular 3-, 4-, 5-, 6- and 7-Cayley graphs, respectively.

Automorphism Groups of a Class of Cubic Cayley Graphs on Symmetric Groups

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.

Heptavalent Symmetric Graphs of Order 16p

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. We classify connected heptavalent symmetric graphs of order 16p for each prime p. As a result, there are two such sporadic graphs with p = 3 and 7, and an infinite family of 1-regular normal Cayley graphs on the group [Formula: see text] with 7|(p – 1).