Suborbital Graphs Of The Groups C, D And U And Graphs Whose Automorphism Groups Contain The Permutation Groups C And D

ABSTRACT

Many authors have studied the suborbital graphs of various group actions and their

corresponding properties. This thesis investigates the actions of the cyclic group and the

dihedral group on the diagonals of a regular -gon and the properties of their corresponding

suborbital graphs. In addition, it focuses on the action of the multiplicative group of units

on

, the set of non zero elements in . The properties of the

corresponding suborbital graphs to this action are also investigated. Lastly, graphs whose

autormorphism groups contain the cyclic group and the dihedral group are constructed.

Transitivity of the actions is established using the Cauchy- Frobenius Lemma or the Orbit-

Stabilizer Theorem. Schur’s algorithm is employed to construct all graphs whose groups of

automorphism contain the cyclic group and the dihedral group . It has been shown that,

and acts transitively on the set of diagonals of a regular - gon and the action is imprimitive

when

is not a prime. The cyclic group, , has two self-paired suborbits when

and only one self-paired suborbit when , while the dihedral group has all its

suborbits self-paired. For the two groups it has been shown that the number, , of the connected

components of the suborbital graph is

and its girth is

when , otherwise it is

zero. The action of on

is shown to be transitive if and only if is a prime.

This action is imprimitive on

when and it has suborbits with two of them being

self-paired. The number of connected components of the suborbital graph , corresponding to

the suborbital , is

. Its girth is , where is the order of the element in .

Finally, it has been shown that if is a permutation group acting transitively on a set of

cardinality and the action is of rank , then the number of regular graphs on vertices whose

groups of automorphism contain is;

, where is the number of

self-paired suborbits of . These among other results have been stated as Lemmas and

Theorems.