University of Illinois Urbana-Champaign

Matchings, Connectivity, and Eigenvalues in Regular Graphs

O, Suil

Content Files
O_Suil.pdf

Permalink

https://hdl.handle.net/2142/26034

Description

Title
Matchings, Connectivity, and Eigenvalues in Regular Graphs
Author(s)
O, Suil
Issue Date
2011-08-25T22:10:01Z
Director of Research (if dissertation) or Advisor (if thesis)
West, Douglas B.
Doctoral Committee Chair(s)
Kostochka, Alexandr V.
Committee Member(s)
  • West, Douglas B.
  • Furedi, Zoltan
  • Yong, Alexander
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2011-08-25T22:10:01Z
Keyword(s)
  • Matching
  • Connectivity
  • Edge-connectivity
  • Eigenvalue
  • Regular graph
  • Postman
  • Path cover
  • Average (edge)-connectivity
  • Total Domination
  • Balloon
  • $r$-dynamic coloring
Abstract
We study extremal and structural problems in regular graphs involving various parameters. In Chapter 2, we obtain the best lower bound for the matching number over $n$-vertex connected regular graphs in terms of edge-connectedness and determine when the matching number is minimized. We also establish the best upper bound for the number of cut-edges over $n$-vertex connected odd regular graphs and determine when the number of cut-edges is maximized. In addition, there is a relationship between the matching number and the total domination number in regular graphs. In Chapter 3, we explore the relationship between eigenvalue and matching number in regular graphs. We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a $l$-edge-connected $d$-regular graph, when $l\leq d-2$. We also study what is the weakest hypothesis on the second largest eigenvalue $\lambda_2$ for a $d$-regular graph $G$ to guarantee that $G$ is $l$-edge-connected. In Chapter 4, we study several extremal problems for regular graphs, including the Chinese postman problem, the path cover number, the average edge-connectivity, and the number of perfect matchings. In Chapter 5, we study an $r$-dynamic coloring problem and give the relationship between the $r$-dynamic chromatic number and the chromatic number in regular graphs. We also study $r$-dynichromatic number of the cartesian product of paths and cycles.
Graduation Semester
2011-08
Permalink
http://hdl.handle.net/2142/26034
Copyright and License Information
Copyright 2011 Suil O

Owning Collections

Graduate Dissertations and Theses at Illinois PRIMARY
Graduate Theses and Dissertations at Illinois
Dissertations and Theses - Mathematics