## LSU Historical Dissertations and Theses

#### Title

Graphs and Number Theory.

1997

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

Jurgen Hurrelbrink

#### Abstract

In the 1930's, L. Redei and H. Reichardt used certain matrices to aid in the determination of the structure of ideal class groups of quadratic number fields. This is a classical number theoretic problem which in general presents diffculties. Ideal class groups are finite abelian groups, and it is a result of Gauss that allows us to determine their 2-rank, in other words the number of cyclic factors of even order. Redei and Reichardt worked on determining the 4-rank, the number of factors of order divisible by 4. Later, the classical study of circulant graphs was utilized to further help this determination. In particular, if we relate a certain circulant graph G to a quadratic number field, then the number of Eulerian Vertex Decompositions of G is closely related to the 4-rank of the ideal class group of the quadratic number field. Circulant graphs however become large rather quickly. Recently, P. E. Conner and J. Hurrelbrink developed the concept of quotient graphs. These are significantly smaller graphs, yet by analyzing their structure, one can determine much of the same number theoretic information, including the 4-rank of the ideal class group of the related quadratic number field, as one can from the underlying circulant graph. Formal quotient graphs are a generalization of quotient graphs and are a useful tool in determining how many graphs on a given number of vertices can be realized as quotient graphs. In Chapter 1, we develop the background information on circulant graphs and explore their structure. We then utilize circulant graphs in Chapter 2 with the development of quotient graphs. In this chapter we determine exactly which graphs on 2, 3, 4, 5 and 7 vertices are quotient graphs. Finally in Chapter 3, we develop the concept of formal quotient graphs as a generalization of quotient graphs. By analyzing the general situation, we are able to count how many formal quotient graphs there are on 11, 13, 17 and 19 vertices and realize many of these graphs as actual quotient graphs.

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