Cayley Graphs Of Dihedral Groups Research Articles

Discrete-time quantum walks on Cayley graphs of Dihedral groups using generalized Grover coins

Discrete-time quantum walks on Cayley graphs of Dihedral groups using generalized Grover coins

Power Cayley Graphs of Dihedral Groups with Certain Order

Combination of the concepts of power graph and Cayley graph associated to groups has led to the introduction to two new variations of Cayley graph known as the union power Cayley graph and the intersection power Cayley graph. The set of vertices for both graphs consist of the elements of a finite group G. Consider any inverse-closed subset S of G, two vertices x and y are adjacent in the union power Cayley graph if 〖xy〗^(-1)∈S or if either one is an integral power of the other. Furthermore, x and y are adjacent in the intersection power Cayley graph if 〖xy〗^(-1)∈S and if either one is an integral power of the other. In this paper, the generalization of the union power Cayley graphs and the intersection power Cayley graphs of the dihedral groups with order 2n, for n≥3 and n=P^m;P is prime and m is a natural number, relative to a specific subset containing rotation elements in the groups is found. In addition, properties of these graphs including the clique numbers, vertex chromatic numbers, girths and diameters are computed. Finally, the characteristics of the graphs, whether they are connected, regular, complete, and planar are also determined.

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.

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.

Cayley graphs of dihedral groups having perfect edge state transfer

Perfect state transfer plays a crucial role in quantum information processing and quantum computation. However, perfect state transfer is a rare phenomenon. As a generalization of perfect state transfer, perfect edge state transfer overcomes this disadvantage and there are more graphs admitting perfect edge state transfer than admitting perfect state transfer. In this paper, we reveal the relations between perfect state transfer and perfect edge state transfer on Cayley graphs. Furthermore, we present necessary and sufficient conditions for the existence of perfect edge state transfer on Cayley graphs of dihedral groups. With these conditions, several concrete constructions of Cayley graphs admitting perfect edge state transfer are proposed. Notably, this is the first attempt at perfect edge state transfer on Cayley graphs.

Automorphism Groups of Cubic Cayley Graphs of Dihedral Groups of Order 2<sup>n</sup><i>p</i><sup>m</sup> (<i>n</i> ≥ 2 and <i>p</i> Odd Prime)

Automorphism Groups of Cubic Cayley Graphs of Dihedral Groups of Order 2npm (n ≥ 2 and p Odd Prime)

On tetravalent symmetric dihedrants

Let Γ be a tetravalent X-arc-transitive Cayley graph of dihedral group for X ≤ AutΓ. Let Xv be the stabilizer of X on v ∈ VΓ. Γ has been determined when it is 2-arc-transitive or one-regular. This paper studies the case where Γ is one-transitive but not one-regular, and gives it an exactly characterization. As an application of this result, we give a compete classification of such graphs when |Xv| ≤ 24. By production, a compete classification is given for the stabilizers of tetravalent symmetric Cayley graphs whenever its order is less than 25.

Diameter 2 Cayley graphs of dihedral groups

We consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a graph with diameter 2 and degree d. We completely determine the asymptotic behaviour of this class of graphs by showing that both limits are asymptotically d2/2.

Locally primitive Cayley graphs of dihedral groups

A graph Γ is called locally-primitive if the vertex stabilizer (AutΓ)α is primitive on the neighbor set of α for each vertex α. In this paper, we classify locally-primitive Cayley graphs of dihedral groups, while 2-arc-transitive Cayley graphs of dihedral groups have been classified by a series of papers in the literature.

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).

Reliability Analysis of the Cayley Graphs of Dihedral Groups

Cayley graphs have many good properties as models of communication networks. This study analyzes the reliability of the Cayley graph based on the dihedral graph. Graph theory and analyses show that almost all Cayley graphs of the dihedral graph D 2 n are optimal super-λ. The number N i(G) of cutsets of size i,λ≤ i≤λ′ is given as N i ( G ) = n ( ( G − 1 ) i − δ ) .

Edge-transitive regular [formula omitted]-covers of the Heawood graph

A regular cover of a graph is said to be an edge-transitive cover if the fibre-preserving automorphism subgroup acts edge-transitively on the covering graph. In this paper we classify edge-transitive regular Z n -covers of the Heawood graph, and obtain a new infinite family of one-regular cubic graphs. Also, as an application of the classification of edge-transitive regular Z n -covers of the Heawood graph, we prove that any bipartite edge-transitive cubic graph of order 14 p is isomorphic to a normal Cayley graph of dihedral group if the prime p > 13 .

Classification of 2-arc-transitive dihedrants

A complete classification of 2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in [D. Marušič, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162–196]. The list consists of the following graphs: cycles C2n, n⩾3; complete graphs K2n, n⩾3; complete bipartite graphs Kn,n, n⩾3; complete bipartite graphs minus a matching Kn,n−nK2, n⩾3; incidence and nonincidence graphs B(H11) and B′(H11) of the Hadamard design on 11 points; incidence and nonincidence graphs B(PG(d,q)) and B′(PG(d,q)), with d⩾2 and q a prime power, of projective spaces; and an infinite family of regular Zd-covers Kq+12d of Kq+1,q+1−(q+1)K2, where q⩾3 is an odd prime power and d is a divisor of q−12 and q−1, respectively, depending on whether q≡1(mod4) or q≡3(mod4), obtained by identifying the vertex set of the base graph with two copies of the projective line PG(1,q), where the missing matching consists of all pairs of the form [i,i′], i∈PG(1,q), and the edge [i,j′] carries trivial voltage if i=∞ or j=∞, and carries voltage h¯∈Zd, the residue class of h∈Z, if and only if i−j=θh, where θ generates the multiplicative group Fq∗ of the Galois field Fq.

Chirality of Toroidal Molecular Graphs

Symmetry properties of a class of toroidal molecular graphs, arising as covers of certain bipartite cubic Cayley graphs of dihedral groups, are studied. Although these symmetries make all vertices and all edges indistinguishable, they imply intrinsic chirality.

On 2-arc-transitivity of Cayley graphs

The classification of 2-arc-transitive Cayley graphs of cyclic groups, given in (J. Algebra. Combin. 5 (1996) 83–86) by Alspach, Conder, Xu and the author, motivates the main theme of this article: the study of 2-arc-transitive Cayley graphs of dihedral groups. First, a previously unknown infinite family of such graphs, arising as covers of certain complete graphs, is presented, leading to an interesting property of Singer cycles in the group PGL(2, q), q an odd prime power, among others. Second, a structural reduction theorem for 2-arc-transitive Cayley graphs of dihedral groups is proved, putting us—modulo a possible existence of such graphs among regular cyclic covers over a small family of certain bipartite graphs—a step away from a complete classification of such graphs. As a byproduct, a partial description of 2-arc-transitive Cayley graphs of abelian groups with at most three involutions is also obtained.

Induced cayley graphs and d0ouble covers

Motivated by a construction of highly expanding simple Cayley graphs of dihedral groups derived from-or induced by-highly expanding Cayley digraphs fo cyclic groups presented by F. R. K. chung, constructions of simple Cayley graphs on a semidirect product of groups and Cayley digraphs of one of its factors are suggested. In case of the non-normal factor being the cyclic group of order 2. a condition is given to derive spectral bounds of the Cayley graph of the product from those of the Cayley graph of the normal factor.