Abstract:

In this thesis we will give an interesting relation between finite rings and their
graphs, such relations are obtained in following way.
Consider a directed graph
on a finite ring
, where
are sets of vertices and edges respectively, and
defined by
. Since is finite, it has an integer characteristic
. If is not a prime, then has zero divisors and is not a unique
factorization ring, but if it is prime, then nevertheless could have zerodivisors
(e.g.,
). Let and be relatively prime numbers, such that
, ! and
define two maps
" #,
$
by "
%&
and
%&
respectively, so " and are homomorphism
maps, suppose that
'() *+,. / + is a directed cycle of length . in a
directed graph , then many interesting algebraic relations will exist between longest
cycles in
,
# and
$, which will be shown up in the chapter III. 