Undirected Cayley Graph Research Articles

Simulations of Cayley graphs of dihedral group

Let Γ be a finite group with identity element e and let S ⊆ Γ − {e} which is inverse-closed, i.e., S = S−1 := {s−1 : s ∈ S}. An undirected Cayley graph on a group Γ with connection set S, denoted by Cay(Γ, S), is a graph with vertex set Γ and edges xy for all pairs x,y ∈ Γ such that xy−1 ∈ S. The dihedral group of order 2n, denoted by D2n, is a group generated by two elements a and b subject to the three relations: an = e, b2 = e, and bab−1 = a−1. In this paper, we investigate the Cayley graphs of the dihedral group for some connection sets S satisfying |S| = 1, 2, 3 by doing simulations using Wolfram Mathematica.

On Generators and Undirected Cayley graphs of Γ_1-Non-Deranged Permutation Groups

A cayley graph of a group , where S is the set of generators of is a graph that has a vertex for each element of and edge for and . In this work, we study the structural properties of undirected cayley graphs of the -non deranged permutation group ( ) obtained from a modulo p function (for and is prime), and established that the generators of are not unique and also the undirected cayley graphs is not simple.

On integral mixed Cayley graphs over non-abelian finite groups admitting an abelian subgroup of index 2

Recently, several works by a number of authors have provided characterizations of integral undirected Cayley graphs over generalized dihedral groups and generalized dicyclic groups. We generalize and unify these results in two different ways. Firstly, we work over arbitrary non-abelian finite groups admitting an abelian subgroup of index 2. Secondly, our main result actually characterizes integral mixed Cayley graphs over such finite groups, in the spirit of a very recent result of Kadyan–Bhattarcharjya in the abelian case.

The Cayley sum graph of ideals of a semigroup

Let S be a regular semigroup, I(S) be the set of ideals of S and M be a subset of I(S). In this paper, we introduce an undirected Cayley graph of S, denoted by Гs,m with elements of I(S) as the vertex set, and, for two distinct vertices I and J, I is adjacent to J if and only if there is an element K of M such that IK = J or JK = I. We study some basic properties of the graph Гs,m such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of Гs,m.

The structure of Cayley graphs of dihedral groups of Valencies 1, 2 and 3

Let G be a group and S be a subset of G such that e ∉ S and S−1 ⊆ S. Then Cay(G, S) is a simple undirected Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy−1 ∈ S. The size of subset S is called the valency of Cay(G, S). In this paper, we determined the structure of all Cay(D2n, S), where D2n is a dihedral group of order 2n, n ≥ 3 and |S| = 1, 2 or 3.

On the asymptotic enumeration of Cayley graphs

In this paper, we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group R has a nontrivial proper normal subgroup N with the property that the automorphism group of the digraph fixes each N-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.

Cops and robbers on directed and undirected abelian Cayley graphs

We show that the cop number of directed and undirected Cayley graphs on abelian groups is in O(n), where n is the number of vertices, by introducing a refined inductive method. With our method, we improve the previous upper bound on cop number for undirected Cayley graphs on abelian groups, and we establish an upper bound on the cop number of directed Cayley graphs on abelian groups. We also use Cayley graphs on abelian groups to construct new Meyniel extremal families, which contain graphs of every order n with cop number in Θ(n).

Diameters of random Cayley graphs of finite nilpotent groups

Abstract We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.

Cayley graphs with few automorphisms

We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above Cayley directed graph does not have bigons. On the other hand, if G is neither generalized dicyclic nor abelian and has an element of order at least $(2rank(G))^{36}$, then it admits an undirected Cayley graph with automorphism group equal to G. This extends classical results for finite groups and free products of groups. The above results are obtained as corollaries of a stronger form of rigidity which says that the rigidity of the graph can be observed in a ball of radius 1 around a vertex. This strong rigidity result also implies that the Cayley (di)graph covers very few (di)graphs. In particular, we obtain Cayley graphs of Tarski monsters which essentially do not cover other quasi-transitive graphs. We also show that a finitely generated group admits a locally finite labelled unoriented Cayley graph with automorphism group equal to itself if and only if it is neither generalized dicyclic nor abelian with an element of order greater than 2.

Navigating directed Cayley graphs of small diameter: A potent Solovay–Kitaev procedure

Let [Formula: see text] be a group and [Formula: see text] be a descending sequence of finite-index normal subgroups. We establish explicit upper bounds on the diameters of the directed Cayley graphs of the [Formula: see text], under some natural hypotheses on the behavior of power and commutator words in [Formula: see text]. The bounds we obtain do not depend on a choice of generating set. Moreover, under reasonable conditions our method provides a fast algorithm for navigating directed Cayley graphs. The proof is closely analogous to the Solovay–Kitaev procedure, which only uses commutator words, but also only constructs small-diameter undirected Cayley graphs. We apply our procedure to give new directed diameter bounds on finite quotients of a large class of regular branch groups, and of [Formula: see text] (for [Formula: see text] even).

Automorphism groups of some generalized Cayley graphs

Let R be a commutative ring with identity element. Graph \(\Gamma ^{n}_{R}\) is defined with vertex set \(R^{n} \setminus \lbrace 0 \rbrace \) and two distinct vertices X and Y are adjacent if and only if there exists an \(n \times n\) lower triangular matrix A with non-zero diagonal entries such that \(AX^T=Y^T\) or \(AY^T=X^T\). By \(B^{T}\), we mean transpose of matrix B. If R is a semigroup with respect to multiplication and \(n=1,\) then \(\Gamma ^{1}_{R}\) is the undirected Cayley graph. In this paper, a prime number p, we find the clique number and automorphism group of \(\Gamma ^{n}_{R},\) where \(R=\mathbb {Z}_{p^2}\) or \(R=\mathbb {Z}_{p^3}\).

Resistance distances in Cayley graphs on symmetric groups

The present paper is devoted to estimating effective resistances in large weighted undirected Cayley graph Cay(Sn,Tn) on symmetric group Sn. We obtain asymptotically exact bounds for minimal resistances in Cay(Sn,Tn). Maximal and average resistances in Cayley graphs on symmetric groups are given in terms of degree, girth and spectral gap. Sharp asymptotic bounds for maximal and average resistances are obtained for star and transposition Cayley graphs. Finally, we find improved estimates of effective resistances in bubble-sort Cayley graphs.

Groups all of whose undirected Cayley graphs are determined by their spectra

The adjacency spectrum [Formula: see text] of a graph [Formula: see text] is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph [Formula: see text] is “determined by its spectrum” (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group [Formula: see text] is Cay-DS if every two cospectral Cayley graphs of [Formula: see text] are isomorphic. In this paper, we study finite DS groups and finite Cay-DS groups. In particular we prove that a finite DS group is solvable, and every non-cyclic Sylow subgroup of a finite DS group is of order [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. We also give several infinite families of non-Cay-DS solvable groups. In particular we prove that there exist two cospectral non-isomorphic [Formula: see text]-regular Cayley graphs on the dihedral group of order [Formula: see text] for any prime [Formula: see text].

On finite groups all of whose cubic Cayley graphs are integral

For any positive integer [Formula: see text], let [Formula: see text] denote the set of finite groups [Formula: see text] such that all Cayley graphs [Formula: see text] are integral whenever [Formula: see text]. Estélyi and Kovács [On groups all of whose undirected Cayley graphs of bounded valency are integral, Electron. J. Combin. 21 (2014) #P4.45.] classified [Formula: see text] for each [Formula: see text]. In this paper, we characterize the finite groups each of whose cubic Cayley graphs is integral. Moreover, the class [Formula: see text] is characterized. As an application, the classification of [Formula: see text] is obtained again, where [Formula: see text].

Rainbow tetrahedra in Cayley graphs

Let Γn be the complete undirected Cayley graph of the odd cyclic group Zn. Connected graphs whose vertices are rainbow tetrahedra in Γn are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs G of largest degree 6, asymptotic diameter |V (G)| and almost all vertices with degree: (a) 6 in G; (b) 4 in exactly six connected subgraphs of the (3, 6, 3, 6)-semi-regular tessellation; and (c) 3 in exactly four connected subgraphs of the {6, 3}regular hexagonal tessellation. These vertices have as closed neighborhoods the union (in a fixed way) of closed neighborhoods in the ten respective resulting tessellations.

On Groups all of whose Undirected Cayley Graphs of Bounded Valency are Integral

A finite group $G$ is called Cayley integral if all undirected Cayley graphs over $G$ are integral, i.e., all eigenvalues of the graphs are integers. The Cayley integral groups have been determined by Kloster and Sander in the abelian case, and by Abdollahi and Jazaeri, and independently by Ahmady, Bell and Mohar in the non-abelian case. In this paper we generalize this class of groups by introducing the class $\mathcal{G}_k$ of finite groups $G$ for which all graphs $\mathrm{Cay}(G,S)$ are integral if $|S| \le k$. It will be proved that $\mathcal{G}_k$ consists of the Cayley integral groups if $k \ge 6;$ and the classes $\mathcal{G}_4$ and $\mathcal{G}_5$ are equal, and consist of: (1) the Cayley integral groups, (2) the generalized dicyclic groups $Dic(E_{3^n} \times \mathbb{Z}_6),$ where $n \ge 1$.

Ends for subsemigroups of finite index

In this paper we study ends of finitely generated semigroups. The ends we are working with are the ends of the undirected Cayley graphs of finitely generated semigroups. We prove that the number of ends is preserved for subsemigroups of finite Rees index, and prove the same result for finite Green index subsemigroups of cancellative semigroups.

Hamiltonian cycles in Cayley graphs of imprimitive complex reflection groups

Generalizing a result of Conway, Sloane, and Wilkes (1989) for real reflection groups, we show the Cayley graph of an imprimitive complex reflection group with respect to standard generating reflections has a Hamiltonian cycle. This is consistent with the long-standing conjecture that for every finite group, G, and every set of generators, S, of G the undirected Cayley graph of G with respect to S has a Hamiltonian cycle.

Groups all of whose undirected Cayley graphs are integral

Let G be a finite group, S⊆G∖{1} be a set such that if a∈S, then a−1∈S, where 1 denotes the identity element of G. The undirected Cayley graph Cay(G,S) of G over the set S is the graph whose vertex set is G and two vertices a and b are adjacent whenever ab−1∈S. The adjacency spectrum of a graph is the multiset of all eigenvalues of the adjacency matrix of the graph. A graph is called integral whenever all adjacency spectrum elements are integers. Following Klotz and Sander, we call a group G Cayley integral whenever all undirected Cayley graphs over G are integral. Finite abelian Cayley integral groups are classified by Klotz and Sander as finite abelian groups of exponent dividing 4 or 6. Klotz and Sander have proposed the determination of all non-abelian Cayley integral groups. In this paper we complete the classification of finite Cayley integral groups by proving that finite non-abelian Cayley integral groups are the symmetric group S3 of degree 3, C3⋊C4 and Q8×C2n for some integer n≥0, where Q8 is the quaternion group of order 8.

CAYLEY GRAPHS OF PARTIALLY ORDERED SETS

In this paper, we introduce the Cayley graph of a partially ordered set (poset). Let (P, ≤) be a poset, and let S be a subset of P. We define the undirected Cayley graph of P, denoted by Cay (P, S), as a graph with vertex-set P and edge-set E consisting of those sets {x, y} such that y ∈ {x, s}ℓ or x ∈ {y, s}ℓ for some s ∈ S, where for a subset T of P, Tℓ is the set of all x ∈ P such that x ≤ t, for all t ∈ T. We study some basic properties of Cay (P, S) such as connectivity, diameter and girth.