#### Abstract

A simple n-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers 1 through n such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex labelings for new families of graphs, all of which are special cases of Seoud and Youssef's conjecture that all unicyclic graphs have a prime labeling.

#### Author Bio

Nathan Diefenderfer is a junior at Northern Arizona University. He is double majoring in mathematics and physics, and expects to go to graduate school in physics after graduation. While he is unsure of what specialty, he is primarily interested in cosmology. In his spare time, he enjoys cooking and baking.

Michael Hastings is a senior at Northern Arizona University. He has been doing research in the department of mathematics for two years on two separate projects. The first was related to factorizing Temperley-Lieb diagram algebras. He worked with another undergraduate student for that project. The second project is the focus of this paper, On Prime Labelings. He worked with six other students and two advisors to come up with a lot of new results. He will be graduating with a B.S. in mathematics and physics in December of 2015. After graduation he is planning to go to graduate school to receive his Ph.D. in physics. His career aspirations include astronaut, optical engineer and university professor.

Levi Heath is a mathematics major at Northern Arizona University with an expected graduation date in May 2015. He is studying to become an actuary, has passed Exam P, and will be sitting for Exam FM in April 2015. His other academic interests include algebra and graph theory.

Hannah Prawzinsky is a junior at Northern Arizona University. She will graduate in May of 2016 with B.S. and B.S.Ed degrees in mathematics. She plans to attend graduate school in mathematics in the fall of 2016. In her spare time, she enjoys oil painting and watching movies.

Briahna Preston is a senior mathematics major at Northern Arizona University. She is quite fond of her dog Kimbo, a 4-year-old male pit bull. Upon graduating in May of 2015, she plans on pursuing a graduate degree in mathematics at a university to be determined.

Emily White is a senior studying mathematics at Northern Arizona University. She began doing math research in the fall of 2014 with her professor Jeff Rushall and Dana Ernst, through the Center for undergraduate Research in Mathematics. She has presented some of this research at the Nebraska Women’s Conference.

Alyssa Whittemore is a senior mathematics major at Northern Arizona University. She has been participating in research in graph theory since her fourth semester and she plans to go to graduate school for a PhD in mathematics. When she has free time, she enjoys dancing, photography, and writing poetry.

#### Recommended Citation

Diefenderfer, Nathan; Hastings, Michael; Heath, Levi N.; Prawzinsky, Hannah; Preston, Briahna; White, Emily; and Whittemore, Alyssa
(2015)
"Prime Vertex Labelings of Families of Unicyclic Graphs,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 16
:
Iss.
1
, Article 14.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol16/iss1/14