The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every nontrivial distance-regular Cayley graph on a dihedral group...

A regular covering projection is called dihedral or abelian if the covering transformation group is dihedral or abelian. A lot of work has been done with regard to the classification of arc-transitive abelian (or elementary abelian, or cyclic) covers of symmetric graphs. In this paper, we investigate arc-transitive dihedral regular covers of symmetric (arc-transitive) cubic graphs. In particula...

Let $\mathbb Z_n$ denote the cyclic group of order $n$. We show how determinant for $G= \mathbb Z_n \times H$ can be simply written in terms $H$. use this to get a complete description integer determinants Z_2 D_8$ where $D_8$ is dihedral $8$, and Q_8$ $Q_8$ quaternion $8$.

In this article, we generalize the concept of group-graded modules by introducing G-weak graded R-modules, which are R-modules a set G left coset representatives, where R is ring. Moreover, prove some properties these modules. Finally, results related to fields and their vector spaces deduced. Many considerable examples provided with more emphasis on symmetric group S3 dihedral D6, gives symmet...

Recently, several works by a number of authors have studied integrality, distance and powers Cayley graphs over some finite groups, such as dicyclic groups (generalized) dihedral groups. Our aim is to generalize and/or give analogues these results for generalized For example, we necessary sufficient condition graph group be integral (i.e., all eigenvalues its adjacency matrix are in Z). We also...

In this paper we introduce a Group Key Management protocol following the idea of classical that extends well-known Diffie–Hellman key agreement to group users. The is defined in non-commutative setting, more precisely, twisted dihedral ring. for an arbitrary cocycle, extending previous agreements considered two main objective work show there no lack security derived from fact larger amount publ...

We construct a new infinite family of four-dimensional isolated symplectic singularities with trivial local fundamental group, answering question Beauville raised in 2000. Three constructions are presented for this family: (1) as blowups the quotient C 4 $\mathbb {C}^4$ by dihedral group order 2 d $2d$ , (2) singular points Calogero–Moser spaces associated groups at equal parameters, and (3) ce...