#### Date Thesis Awarded

4-2022

#### Access Type

Honors Thesis -- Open Access

#### Degree Name

Bachelors of Science (BS)

#### Department

Mathematics

#### Advisor

Charles Johnson

#### Committee Members

Eric Swartz

Weizhen Mao

#### Abstract

A path cover of a tree T is a collection of induced paths of T that are vertex disjoint and cover all the vertices of T. A minimum path cover (MPC) of T is a path cover with the minimum possible number of paths, and that minimum number is called the path cover number of T. A tree can have just one or several MPC's. Prior results have established equality between the path cover number of a tree T and the largest possible multiplicity of an eigenvalue that can occur in a symmetric matrix whose graph is that tree. We hope to gain insights into the different ways that maximum multiplicity occurs among the multiplicity lists of T by enumerating its MPC's. The overall strategy is to divide and conquer. Given any tree T, several techniques are introduced to decompose T into smaller components. Then, the number of MPC's of these smaller trees can be calculated and recombined to obtain the number of MPC's for the original tree T.

#### Recommended Citation

Sher, Merielyn, "The Enumeration of Minimum Path Covers of Trees" (2022). *Undergraduate Honors Theses.* William & Mary. Paper 1809.

https://scholarworks.wm.edu/honorstheses/1809