M.Sc. / 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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uj/uj:3247 |
| Date | 27 August 2012 |
| Creators | Swarts, Jacobus Stephanus |
| Source Sets | South African National ETD Portal |
| Detected Language | English |
| Type | Thesis |
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