#### Document Type

Dissertation

#### Publication Date

5-2022

#### First Advisor

Joshua Holden

#### Abstract

The tools of graph theory can be used to investigate the structure

imposed on the integers by various relations. Here we investigate two

kinds of graphs. The first, a square product graph, takes for its vertices

the integers 1 through n, and draws edges between numbers whose product

is a square. The second, a square product graph, has the same vertex set,

and draws edges between numbers whose sum is a square.

We investigate the structure of these graphs. For square product

graphs, we provide a rather complete characterization of their structure as

a union of disjoint complete graphs. For square sum graphs, we investigate

some properties such as degrees of vertices, connectedness, hamiltonicity,

and planarity.

#### Recommended Citation

Trent, Lee, "Structure of Number Theoretic Graphs" (2022). *Mathematical Sciences Technical Reports (MSTR)*. 179.

https://scholar.rose-hulman.edu/math_mstr/179

#### Included in

Applied Mathematics Commons, Discrete Mathematics and Combinatorics Commons, Number Theory Commons

## Comments

Senior Thesis, Department of Mathematics, Rose-Hulman Institute of Technology