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Abstract:
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This dissertation examines various properties of commutative rings and modules
using algebraic combinatorial methods. If the graph is properly associated to a ring
R or to an R-module M, then examination of its properties gives useful information
about the ring R or R-module M.
This thesis discusses the determination of the radius of the total graph of a
commutative ring R in the case when this graph is connected. Typical extensions
such as polynomial rings, formal power series, idealization of the R-module M and
relations between the total graph of the ring R and its extensions are also dealt
with.
The total graph of a module, a generalization of the total graph of a ring is
presented. Various properties are proved and some relations to the total graph of a
ring as well as to the zero-divisor graph are established.
To gain a better understanding of clean rings and their relatives, the clean graph
C¡(R) of a commutative ring with identity is introduced and its various proper-
ties established. Further investigation of clean graphs leads to additional results
concerning other classes of commutative rings.
One of the topics of this thesis is the investigation of the properties of the cor-
responding line graph L(T¡(R)) of the total graph T¡(R). The classi¯cation of
all commutative rings whose line graphs of the total graph are planar or toroidal
is given. It is shown that for every integer g ¸ 0 there are only ¯nitely many
commutative rings such that °(L(T¡(R))) = g.
Also, in this thesis all toroidal graphs which are intersection graphs of ideals
of a commutative ring R are classi¯ed. An improvement over the previous results
concerning the planarity of these graphs is presented. |