(University of the Free State, 2016-09) De Klerk, Ben-Eben; Meyer, J. H.
English: Given a set S, and any selfmap ƒ: S→S, the functional graph associated
with ƒ can be described as a graph with vertex set S and directed edge set
E = {(u; v) ϵ S2 : ƒ (u) = v}. A classification of all functional graphs induced
by lattice endomorphisms has recently been done by J. Szigeti ([12]). In this
dissertation, we aim to achieve a similar type of classi_cation with respect to
functional graphs induced by endomorphisms on certain abelian groups.
A method for finding all functional graphs that can be induced by endomorphisms
of a group has been developed for all groups of the form Znp
with p any
prime, n ϵ N, and Zn for any n ϵ N, as well as all cyclic groups.
A deep connection between the functional graphs corresponding to group endomorphisms
and the minimal polynomial of the matrix representation of the
group endomorphism has been found.