Undergraduate Honors Thesis Projects

Date of Award

2016

Document Type

Honors Paper

Degree Name

Mathematics-BS

Department

Mathematical Sciences

Advisor

Jeremy Moore

First Committee Member

Jeremy Moore

Second Committee Member

Ryan Berndt

Third Committee Member

Jonathan DeCoster

Keywords

graph theory, minimal edges, automorphism group, minimal vertices, dicyclic

Subject Categories

Discrete Mathematics and Combinatorics

Abstract

We find the smallest degree of a graph with automorphism group isomorphic to the dicyclic group with 4n elements, denoted α(Dic_n). We also find the fewest edges a minimum-order graph with dicyclic automorphism group and a minimal number of vertices can have. For n not a power of 2, the value of α(Dic_n) is significantly less than the best previously known upper bound, 8n. Such an edge-minimized vertex-minimal graph is constructed and shown to have automorphism group Dic_n . That the exhibited graph is minimal is verified using a combination of techniques similar to those developed previously for Abelian groups and earlier results for the special case Dic_n (where n is a power of 2).

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