We give a non-deterministic algorithm that expresses elements of SL_N(Z), for N > 2, as words in a finite set of generators, with the length of these words at most a constant times the word metric. We show that the non-deterministic time-complexity of the subtractive version of Euclid's algorithm for finding the greatest common divisor of N > 2 integers a_1,..., a_N is at most a constant times N log n where n := max {|a_1|,..., |a_N|}. This leads to an elementary proof that for N > 2 the word metric in SL_N(Z) is biLipschitz equivalent to the logarithm of the matrix norm -- an instance of a theorem of Mozes, Lubotzky and Raghunathan. And we show constructively that there exists K>0 such that for all N > 2 and primes p, the diameter of the Cayley graph of SL_N(F_p) with respect to the generating set {e_{ij} \mid i \neq j} is at most K N^2 \log p.
This paper describes a non-deterministic polynomial-time algorithm to find a path of length O(log p loglog p) between any two vertices of the Cayley graph of SL(2,Z/pZ).
Cayley hash functions are based on a simple idea of using a pair of (semi)group elements, $A$ and $B$, to hash the 0 and 1 bit, respectively, and then to hash an arbitrary bit string in the natural way, by using multiplication of elements in the (semi)group. In this paper, we focus on hashing with $2 \times 2$ matrices over $F_p$. Since there are many known pairs of $2 \times 2$ matrices over $Z$ that generate a free monoid, this yields numerous pairs of matrices over $F_p$, for a sufficiently large prime $p$, that are candidates for collision-resistant hashing. However, this trick can "backfi…
In this paper, generalized Cayley graphs are studied. It is proved that every generalized Cayley graph of order 2p is a Cayley graph, where p is a prime. Special attention is given to generalized Cayley graphs on Abelian groups. It is proved that every generalized Cayley graph on an Abelian group with respect to an automorphism which acts as inversion is a Cayley graph if and only if the group is elementary Abelian 2-group, or its Sylow 2-subgroup is cyclic. Necessary and sufficient conditions for a generalized Cayley graph to be unworthy are given.
We classify the planar cubic Cayley graphs of connectivity 2, providing an explicit presentation and embedding for each of them. Combined with [9] this yields a complete description of all planar cubic Cayley graphs.
As a main result of this paper we give conditions under which the generalized $X$-join of Cayley graphs is a Cayley graph. In particular, we show that $X$-join of isomorphic Cayley graphs is a Cayley grpah. To do this, new properties for a generalized wreath product of permutation groups are given in the case where the base group acts regularly. These are used to give conditions for the generalized wreath product to contain a regular subgroup, which are then applied to generalized $X$-joins of Cayley graphs. Along the way, it is shown that the generalized $X$-join of isomorphic Cayley graphs w…
We solve two open problems regarding the classification of certain classes of Cayley graphs with integer eigenvalues. We first classify all finite groups that have a "non-trivial" Cayley graph with integer eigenvalues, thus solving a problem proposed by Abdollahi and Jazaeri. The notion of Cayley integral groups was introduced by Klotz and Sander. These are groups for which every Cayley graph has only integer eigenvalues. In the second part of the paper, all Cayley integral groups are determined.
We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a spanning tree in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this subtree is a Hamiltonian cycle, but convergence is meant in a stronger sense. These latter are related to whether having a Hamiltonian cycle is a testable graph property.
In this paper, we consider the problem on the existence of perfect state transfer(PST for short) on semi-Cayley graphs over abelian groups (which are not necessarily regular), i.e on the graphs having semiregular and abelian subgroups of automorphisms with two orbits of equal size. We stablish a characterization of semi-Cayley graphs over abelian groups having PST. As a result, we give a characterization of Cayley graphs over groups with an abelian subgroup of index 2 having PST, which improves the earlier results on Cayley graphs over abelian groups, dihedral groups and dicyclic group and det…
We introduce a family of graphs that generalises the class of Cayley graphs. For non-empty subsets L, R of a group G, the two-sided Cayley graph 2SC(G;L,R) is the directed graph with vertex set G and an arc from x to y if and only if y=a^{-1}xb for some a in L and b in R. Thus, in common with Cayley graphs, two-sided Cayley graphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine when two-sided Cayley graphs are simple undirected graphs, and give sufficient conditions for them to be connected, vertex-transitive, or Cayl…
We consider the Cayley graph ${\rm C}(W,S)$ of a Coxeter system $(W,S)$ and describe all maximal $2$-cliques in this graph, i.e. maximal subsets in the vertex set such that the distance between any two distinct elements is equal to $2$. As an application, we show that every automorphism of the half of Cayley graph is uniquely extendable to an automorphism of the Cayley graph if $|S|\ge 5$.
We prove that a Cayley graph can be embedded in the euclidean plane without accumulation points of vertices if and only if it is the 1-skeleton of a Cayley complex that can be embedded in the plane after removing redundant simplices. We also give a characterisation of these Cayley graphs in term of group presentations, and deduce that they can be effectively enumerated.
We construct a connected cubic nonnormal Cayley graph on $\mathrm{A}_{2^m-1}$ for each integer $m\geqslant4$ and determine its full automorphism group. This is the first infinite family of connected cubic nonnormal Cayley graphs on nonabelian simple groups.
We prove that every vertex transitive, planar, 1-ended, graph covers every graph whose balls of radius r are isomorphic to the ball of radius r in G for a sufficiently large r. We ask whether this is a general property of finitely presented Cayley graphs, as well as further related questions.
A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs : abelian groups and generalised dicyclic groups. Indeed, any Cayley graph on such a group admits specific additional graph automorphisms that depend only on the group. Recently, Dobson and the last two authors showed that almost all Cayley graphs on abelian groups admit no automorphisms other than these obvious necessary ones. In this paper, we prove the analogous resul…
In a recent paper (arXiv:1505.01475 ) Estélyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph $G(10,2)$, occurring as the graph $40$ in the Foster census of connected symmetric trivalent graphs.
In this note we prove that all connected Cayley graphs of every finite group $Q \times H$ are 1-factorizable, where $Q$ is any non-trivial group of 2-power order and $H$ is any group of odd order.
Extending earlier results of Godsil and of Dobson and Malnic on Johnson graphs, we characterise those merged Johnson graphs $J=J(n,k)_I$ which are Cayley graphs, that is, which are connected and have a group of automorphisms acting regularly on the vertices. We also characterise the merged Johnson graphs which are not Cayley graphs but which have a transitive group of automorphisms with vertex-stabilisers of order $2$. Even though these merged Johnson graphs are all vertex-transitive, we show that only relatively few of them are Cayley graphs or have a transitive group of automorphisms with ve…
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 $\operatorname{Sym}(G)$). 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.
A Cayley graph of a group $H$ is a finite simple graph $Γ$ such that ${\rm Aut}(Γ)$ contains a subgroup isomorphic to $H$ acting regularly on $V(Γ)$, while a Haar graph of $H$ is a finite simple bipartite graph $Σ$ such that ${\rm Aut}(Σ)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(Σ)$ and the $H$-orbits are equal to the bipartite sets of $Σ$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner…
In this paper we show that two distinct conjectures, the first proposed by Babai and Godsil in $1982$ and the second proposed by Xu in $1998$, concerning the asymptotic enumeration of Cayley graphs are in fact equivalent. This result follows from a more general theorem concerning the asymptotic enumeration of a certain family of Cayley graphs.
We obtain a complete description of the planar cubic Cayley graphs, providing an explicit presentation and embedding for each of them. This turns out to be a rich class, comprising several infinite families. We obtain counterexamples to conjectures of Mohar, Bonnington and Watkins. Our analysis makes the involved graphs accessible to computation, corroborating a conjecture of Droms.
We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.
We obtain an effective enumeration of the family of finitely generated groups admitting a faithful, properly discontinuous action on some 2-manifold contained in the sphere. This is achieved by introducing a type of group presentation capturing exactly these groups. Extending this in a companion paper, we find group presentations capturing the planar finitely generated Cayley graphs. Thus we obtain an effective enumeration of these Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al.
The relative Cayley graph of a group $G$ with respect to its proper subgroup $H$, is a graph whose vertices are elements of $G$ and two vertices $h\in H$ and $g\in G$ are adjacent if $g=hc$ for some $c\in C$, where $C$ is an inversed-closed subset of $G$. We study the relative Cayley graphs and, among other results, we discuss on their connectivity and forbidden structures, and compute some of their important numerical invariants.
We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, Rödl, and Schacht, who treated the abelian case. The proof relies on Grothendieck's inequality. As a corollary, we also prove that a similar result holds in all vertex-transitive graphs.
On arc-transitive inner-automorphic Cayley graphs on dihedral groups
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, MOE Beijing Normal University（北京师范大学数学科学学院、数学与复杂系统教育部重点实验室）；Center for Combinatorics, LPMC Nankai University（南开大学组合数学中心、低维空间拓扑及其应用教育部重点实验室）
Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon--Roichman. His pr…
We show that a group admits a planar, finitely generated Cayley graph if and only if it admits a special kind of group presentation we introduce, called a planar presentation. Planar presentations can be recognised algorithmically. As a consequence, we obtain an effective enumeration of the planar Cayley graphs, yielding in particular an affirmative answer to a question of Droms et al. asking whether the planar groups can be effectively enumerated.
In this paper, rough approximations of Cayley graphs are studied and rough edge Cayley graphs are introduced. Furthermore, a new algebraic definition called pseudo-Cayley graphs containing Cayley graphs is proposed. Rough approximation is expanded to pseudo-Cayley graphs. Also, rough vertex pseudo-Cayley graphs and rough pseudo-Cayley graphs are introduced. Some theorems are provided, form which some properties such as connectivity and optimal connectivity are derived. This approach opens a new research field in sciences such as data networks.
A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly(n).
We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.
In this paper, we give a characterization for a class of edge-transitive Cayley graphs, and provide methods for constructing Cayley graphs with certain symmetry properties. Also this study leads to construct and characterise a new family of half-transitive graphs.
We give an introduction to the Cayley-Abels graph for a totally disconnected, locally compact (tdlc) group. It is a generalization of the Cayley graph. We illustrate that on the one hand, Cayley-Abels graphs are useful tools to extend concepts concerning finitely generated groups to compactly generated, tdlc groups and on the other hand, they can be used to investigate properties that, for finitely generated groups, are trivial.
In this work, we explore edge direction, transitivity, and connectedness of Cayley graphs of gyrogroups. More specifically, we find conditions for a Cayley graph of a gyrogroup to be undirected, transitive, and connected. We also show a relationship between the cosets of a certain type of subgyrogroups and the connected components of Cayley graphs. Some examples regarding these findings are provided.
A C4C8 net is a trivalent decoration made by alternating squares C4 and octagons C8. It can cover either a cylinder or a torus. In this paper we determine rhomboidal C4C8 toris which are Cayley graphs.
A classification is given for factorizations of almost simple groups with at least one factor solvable, and it is then applied to characterize $s$-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary: Except the cycles, every non-bipartite connected 3-arc-transitive Cayley graph of a solvable group is a cover of the Petersen graph or the Hoffman-Singleton graph.
All Cayley representations of the distant graph $Γ_Z$ over integers are characterized as Neumann subgroups of the extended modular group. Possible structures of Neumann subgroups are revealed and it is shown that every such a structure can be realized.
We define the notion of rough Cayley graph for compactly generated locally compact groups in terms of quasi-actions. We construct such a graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to quasi-isometry. A class of examples is given by the Cayley graphs of cocompact lattices in compactly generated groups. As an application, we show that a compactly generated group has polynomial growth if and only if its rough Cayley graph has polynomial growth (same for intermediate and exponential growth). Moreover, a unimodular compactly generated group is…
A diagonal base of a Sylow 2-subgroup $P_n(2)$ of symmetric group $S_{2^n}$ is a minimal generating set of this subgroup consisting of elements with only one non-zero coordinate in the polynomial representation. For different diagonal bases Cayley graphs of $P_n(2)$ may have different girths (i.e. minimal lengths of cycles) and thus be non-isomorphic. In presented paper all possible values of girths of Cayley graphs of $P_n(2)$ on diagonal bases are calculated. A criterion for whenever such Cayley graph has girth equal to 4 is presented. A lower bound for the number of different non-isomorphic…
In this work we characterise Cayley graphs of Coxeter groups with respect to the standard generating set that admit uncountable vertex stabilisers. As a corollary, we fully identify finitely generated Coxeter groups for which the automorphism group of their Cayley graph with respect to the standard generating set is not discrete when equipped with the permutation topology. As an application, we also provide new explicit constructions of vertex-transitive graphs of infinite degree that have locally compact automorphism groups.
We investigate which finite Cayley graphs admit a quantum ergodic eigenbasis, proving that this holds for any Cayley graph on a group of size $n$ for which the sum of the dimensions of its irreducible representations is $o(n)$, yet there exist Cayley graphs that do not have any quantum ergodic eigenbasis.
We show that every non-trivial compact connected group and every non-trivial general or special linear group over an infinite field admits a generating set such that the associated Cayley graph has infinite diameter.
We prove that the value of the critical probability for percolation on an abelian Cayley graph is determined by its local structure. This is a partial positive answer to a conjecture of Schramm: the function pc defined on the set of Cayley graphs of abelian groups of rank at least 2 is continuous for the Benjamini-Schramm topology. The proof involves group-theoretic tools and a new block argument.
