KUKU

KU ScholarWorks

  • myKU
  • Email
  • Enroll & Pay
  • KU Directory
    • Login
    View Item 
    •   KU ScholarWorks
    • Dissertations and Theses
    • Dissertations
    • View Item
    •   KU ScholarWorks
    • Dissertations and Theses
    • Dissertations
    • View Item
    JavaScript is disabled for your browser. Some features of this site may not work without it.

    Graphs of Polytopes

    Thumbnail
    View/Open
    Espenschied_ku_0099D_13679_DATA_1.pdf (562.5Kb)
    Issue Date
    2014-12-31
    Author
    Espenschied, William Joshua
    Publisher
    University of Kansas
    Format
    109 pages
    Type
    Dissertation
    Degree Level
    Ph.D.
    Discipline
    Mathematics
    Rights
    Copyright held by the author.
    Metadata
    Show full item record
    Abstract
    The graph of a polytope is the graph whose vertex set is the set of vertices of the polytope, and whose edge set is the set of edges of the polytope. Several problems concerning graphs of polytopes are discussed. The primary result is a set of bounds (Theorem 39) on the maximal size of an anticlique (sometimes called a coclique, stable set, or independent set) of the graph of a polytope based on its dimension and number of vertices. Two results concerning properties preserved by certain operations on polytopes are presented. The first is that the Gale diagram of a join of polytopes is the direct sum of the Gale diagrams of the polytopes and dually, that the Gale diagram of a direct sum of polytopes is the join of their Gale diagrams (Theorem 23). The second is that if two polytopes satisfy a weakened form of Gale's evenness condition, then so does their product (Theorem 32). It is shown, by other means, that, with only two exceptions, the complete bipartite graphs are never graphs of polytopes (Theorem 47). The techniques developed throughout are then used to show that the complete 3-partite graph K_{1,n,m} is the graph of a polytope if and only if K_{n,m} is the graph of a polytope (Theorem 49). It is then shown that K_{2,2,3} and K_{2,2,4} are never graphs of polytopes. A conjecture is then stated as to precisely when a complete multipartite graph is the graph of a polytope. Finally, a section is devoted to results concerning the dimensions for which the graph of a crosspolytope is the graph of a polytope.
    URI
    http://hdl.handle.net/1808/18668
    Collections
    • Dissertations [4474]
    • Mathematics Dissertations and Theses [179]

    Items in KU ScholarWorks are protected by copyright, with all rights reserved, unless otherwise indicated.


    We want to hear from you! Please share your stories about how Open Access to this item benefits YOU.


    Contact KU ScholarWorks
    785-864-8983
    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    Image Credits
     

     

    Browse

    All of KU ScholarWorksCommunities & CollectionsThis Collection

    My Account

    LoginRegister

    Statistics

    View Usage Statistics

    Contact KU ScholarWorks
    785-864-8983
    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    Image Credits
     

     

    The University of Kansas
      Contact KU ScholarWorks
    Lawrence, KS | Maps
     
    • Academics
    • Admission
    • Alumni
    • Athletics
    • Campuses
    • Giving
    • Jobs

    The University of Kansas prohibits discrimination on the basis of race, color, ethnicity, religion, sex, national origin, age, ancestry, disability, status as a veteran, sexual orientation, marital status, parental status, gender identity, gender expression and genetic information in the University’s programs and activities. The following person has been designated to handle inquiries regarding the non-discrimination policies: Director of the Office of Institutional Opportunity and Access, IOA@ku.edu, 1246 W. Campus Road, Room 153A, Lawrence, KS, 66045, (785)864-6414, 711 TTY.

     Contact KU
    Lawrence, KS | Maps