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.
","abstract_html":"<p>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.</p>","abstract_has_math":false,"creators":["Yalman, Demet"],"institution":null,"degree_name":"Master of Science in Mathematics (M.S.)","degree_level":"Thesis (open access)","degree_discipline":"Department of Mathematical Sciences","degree_department":null,"school":null,"contributors":["Colton Magnant","Goran Lesaja"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-01-01T08:00:00Z","date_published":"2015-01-01T08:00:00Z","updated_at":"2026-07-24T02:28:22Z","subjects":["ETD","Edge-swap heuristic","Dense tree","Minimum spanning tree","Leech tree","Modular Leech tree","Distances between leaves","Discrete Mathematics and Combinatorics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.georgiasouthern.edu/etd/1297","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Colton Magnant","Goran Lesaja"]},{"key":"dc:creator","label":"Author","values":["Yalman, Demet"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2015-06-23T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis (open access)"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science in Mathematics (M.S.)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["ETD","Edge-swap heuristic","Dense tree","Minimum spanning tree","Leech tree","Modular Leech tree","Distances between leaves","Discrete Mathematics and Combinatorics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.georgiasouthern.edu/etd/1297"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Labeled Trees and Spanning Trees: Computational Discrete Mathematics and Applications"]}]}],"canonical_facts":{"dc:contributor":["Colton Magnant","Goran Lesaja"],"dc:creator":["Yalman, Demet"],"dc:date.available":["2015-06-23T07:00:00Z"],"dc:description.abstract":["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.
"],"dc:identifier":["https://digitalcommons.georgiasouthern.edu/etd/1297"],"dc:subject":["ETD","Edge-swap heuristic","Dense tree","Minimum spanning tree","Leech tree","Modular Leech tree","Distances between leaves","Discrete Mathematics and Combinatorics"],"dc:title":["Labeled Trees and Spanning Trees: Computational Discrete Mathematics and Applications"],"thesis:degree_discipline":["Department of Mathematical Sciences"],"thesis:degree_level":["Thesis (open access)"],"thesis:degree_name":["Master of Science in Mathematics (M.S.)"]},"updated_at":"2026-07-24T02:28:22Z"}