Undirected Cayley Graph Research Articles

Generalized Cayley graphs associated to commutative rings

Let R be a commutative ring with identity element. For a natural number n, we associate a simple graph, denoted by ΓRn, with Rn⧹{0} as the vertex set and two distinct vertices X and Y in Rn being adjacent if and only if there exists an n×n lower triangular matrix A over R whose entries on the main diagonal are non-zero and such that AXT=YTor AYT=XT, where, for a matrix B, BT is the matrix transpose of B. When we consider the ring R as a semigroup with respect to multiplication, then ΓR1 is the usual undirected Cayley graph (over a semigroup). Hence ΓRn is a generalization of Cayley graph. In this paper we study some basic properties of ΓRn. We also determine all isomorphic classes of finite commutative rings whose generalized Cayley graph has genus at most three.

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).

Integral Cayley Graphs over Abelian Groups

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

Error graphs and the reconstruction of elements in groups

Packing and covering problems for metric spaces, and graphs in particular, are of essential interest in combinatorics and coding theory. They are formulated in terms of metric balls of vertices. We consider a new problem in graph theory which is also based on the consideration of metric balls of vertices, but which is distinct from the traditional packing and covering problems. This problem is motivated by applications in information transmission when redundancy of messages is not sufficient for their exact reconstruction, and applications in computational biology when one wishes to restore an evolutionary process. It can be defined as the reconstruction, or identification, of an unknown vertex in a given graph from a minimal number of vertices (erroneous or distorted patterns) in a metric ball of a given radius r around the unknown vertex. For this problem it is required to find minimum restrictions for such a reconstruction to be possible and also to find efficient reconstruction algorithms under such minimal restrictions. In this paper we define error graphs and investigate their basic properties. A particular class of error graphs occurs when the vertices of the graph are the elements of a group, and when the path metric is determined by a suitable set of group elements. These are the undirected Cayley graphs. Of particular interest is the transposition Cayley graph on the symmetric group which occurs in connection with the analysis of transpositional mutations in molecular biology [P.A. Pevzner, Computational Molecular Biology: An Algorithmic Approach, MIT Press, Cambridge, MA, 2000; D. Sankoff, N. El-Mabrouk, Genome rearrangement, in: T. Jiang, T. Smith, Y. Xu, M.Q. Zhang (Eds.), Current Topics in Computational Molecular Biology, MIT Press, 2002]. We obtain a complete solution of the above problems for the transposition Cayley graph on the symmetric group.

Covering the alternating groups by products of cycle classes

Given integers k , l ⩾ 2 , where either l is odd or k is even, we denote by n = n ( k , l ) the largest integer such that each element of A n is a product of k cycles of length l. For an odd l, k is the diameter of the undirected Cayley graph Cay ( A n , C l ) , where C l is the set of all l-cycles in A n . We prove that if k ⩾ 2 and l ⩾ 9 is odd and divisible by 3, then 2 3 k l ⩽ n ( k , l ) ⩽ 2 3 k l + 1 . This extends earlier results by Bertram [E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory 12 (1972) 368–380] and Bertram and Herzog [E. Bertram, M. Herzog, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (2001) 87–99].

On the Automorphism Groups of Cayley Graphs of Finite Simple Groups

Let G be a nite nonabelian simple group and let be a connected undirected Cayley graph for G. The possible structures for the full automorphism group Aut are specied. Then, for certain nite simple groups G, a sucient condition is given under which G is a normal subgroup of Aut. Finally, as an application of these results, several new half-transitive graphs are constructed. Some of these involve the sporadic simple groups G =J 1 ,J 4, Ly and BM, while others fall into two innite families and involve the Ree simple groups and alternating groups. The two innite families contain examples of half-transitive graphs of arbitrarily large valency.

Some topological properties of star connected cycles

Star connected cycles are shown to be an undirected Cayley graph, and the graph automorphism group is determined. A routing algorithm is given, which finds an optimal path in polynomial time. The diameter and tight bounds on the average distance are computed.

Computing the Diameter in Multiple-Loop Networks

A multiple-loop network is an undirected Cayley graph of a cyclic group. Algorithms for computing the diameter of multiple-loop networks are discussed. By a geometrical approach the problem of computing the diameter of a multiple-loop network (with k hops) is transformed to an equivalent problem in the integer lattice Z k . In the special case of double-loop networks this leads to an algorithm with running time O(log n). This is an undirected analogue to the result of Cheng and Hwang, who recently found an O(log n) algorithm for computing diameters of directed double-loop networks.

Growth series of some wreath products

The growth series of certain finitely generated groups which are wreath products are investigated. These growth series are intimately related to the traveling salesman problem on certain graphs. A large class of these growth series is shown to consist of irrational algebraic functions. n=0 Since only one generating set will be associated with each group below, the generating set associated with fr(x) will be obvious. Let H be a group with a finite generating set Sh ■ These will be fixed for the rest of the paper. The Cayley graph of the pair (H, Sh) is, as usual, the directed graph whose vertices are the elements of H and there is an edge from a vertex hx Xoa vertex h2 if and only if h2 = hxh for some h in SH US^1. In particular, the edge from hx to h2 has an opposite edge from h2 to hx . Let C be the graph gotten from the Cayley graph of (H, Sh) simply by identifying opposite edges. In other words, C might be called the undirected Cayley graph of (H,SH). hex K also be a group with a finite generating set Sk ■ These will also be fixed for the rest of the paper. It is possible to form what might be called a restricted direct product group P, which consists of all functions p from the vertices of C to K such that there are only finitely many vertices v in C with p(v) t? 1. This is a subgroup of the direct product group whose elements consist of all functions from the vertex set of C to K. The group P admits H as a group of automorphisms by means of the action of H on C. The resulting semidirect product P x H is the restricted wreath product K\H.

On Hamiltonian cycles in Cayley graphs of wreath products

In this paper we will show that a Hamiltonian cycle exists in the undirected Cayley graph of the wreath product, Z m wr Z n , of two cyclic groups if m is even or equal to 3.

On 1-factorizability of Cayley graphs

In this paper we consider the existence of a 1-factorization of undirected Cayley graphs of groups of even order. We show that a 1-factorization exists for all generating sets for even order abelian groups, dihedral groups, and dicyclic groups and for all minimal generating sets for even order nilpotent groups and for D m × Z n . We also derive other results that are useful in considering specific Cayley graphs. These results support the conjecture that all Cayley graphs of groups of even order are 1-factorizable. If this is not the case the same result may hold for minimal generating sets.