Term of Award

Summer 2015

Degree Name

Master of Science in Mathematics (M.S.)

Document Type and Release Option

Thesis (open access)

Copyright Statement / License for Reuse

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.


Department of Mathematical Sciences

Committee Chair

Hua Wang

Committee Member 1

Colton Magnant

Committee Member 2

Goran Lesaja


In this thesis, we examine two topics. In the first part, we consider Leech tree which is a tree of order n with positive integer edge weights such that the weighted distances between pairs of vertices are exactly from 1 to n choose 2. Only five Leech trees are known and some non-existence results have been presented through the years. Variations of Leech trees such as the minimal distinct distance trees and modular Leech trees have been considered in recent years. In this thesis, such Leech-type questions on distances between leaves are studied as well as some other labeling questions related to the original motivation for Leech trees. As a second part, we consider the question of finding spanning trees under various restrictions is studied. A “dense” tree, from graph theoretical point of view, has small total distances between vertices and large number of substructures. In this thesis, the “density” of a spanning tree is conveniently measured by the total distance of the tree. By utilizing established conditions and relations between trees with the minimum total distance, an edge-swap heuristic for generating “dense” spanning trees is presented.