### Abstract:

This thesis is concerned with one possible interplay between commutative algebra and
graph theory. Specifically, we associate with a commutative ring R a graph and then set
out to determine how the ring's properties influence the chromatic and clique numbers of
the graph.
The graph referred to is obtained by letting each ring element be represented by a
vertex in the graph and joining two vertices when the product of their corresponding ring
elements is equal to zero.
The thesis focuses on rings that have a finite chromatic number, where the chromatic
number of the ring is equal to the chromatic number of the associated graph. The nilradical
of the ring plays a prominent role in these- investigations.
Furthermore, the thesis also discusses conditions under which the chromatic and clique
numbers of the associated graph are equal. The thesis ends with a discussion of rings with
low (< 5) chromatic number and an example of a ring with clique number 5 and chromatic
number 6.