Detailing the Connection Between a Family of Polar Graphs and Tremain Equiangular Tight Frames

Author

Nicholas Brown, South Dakota State UniversityFollow

Document Type

Dissertation - Open Access

Award Date

2021

Degree Name

Doctor of Philosophy (PhD)

Department / School

Mathematics and Statistics

First Advisor

John Jasper

Keywords

Finite Fields, Frame Theory, Graph Theory

Abstract

The relationship between strongly regular graphs and equiangular tight frames has been known for several years, and this relationship has been used to construct many of the most recent examples of new strongly regular graphs. In this paper, we present an explicit construction of a family of equiangular tight frames using the geometry of a quadratic space over the field of four elements. We observe that these frames give rise to a strongly regular graph on a subset of points of a quadratic space over the field with 4 elements. We then demonstrate an isomorphism between this graph and a classical construction of polar graphs. While this family of graphs is known to exist, their construction using a Tremain ETF is much simpler, requiring the existence of Steiner triple systems and Hadamard matrices of the appropriate size, whereas the original constructions require computing intersections of hyperplanes.

Library of Congress Subject Headings

Graph theory.
Frames (Vector analysis)

Number of Pages

85

Publisher

South Dakota State University

Recommended Citation

Brown, Nicholas, "Detailing the Connection Between a Family of Polar Graphs and Tremain Equiangular Tight Frames" (2021). Electronic Theses and Dissertations. 211.
https://openprairie.sdstate.edu/etd2/211