Optimal Tilings of Bipartite Graphs Using Self-Assembling DNA

Eric Redmon; Miles Mena; Megan Vesta; Alvi Renzyl Cortes; Lauren Gernes; Simon Merheb; Nick Soto; Chandler Stimpert; Amanda Harsy

Eric Redmon Lewis University
Miles Mena Lewis University
Megan Vesta Lewis University
Alvi Renzyl Cortes Lewis University
Lauren Gernes Lewis University
Simon Merheb Lewis University
Nick Soto Lewis University
Chandler Stimpert Lewis University
Amanda Harsy Lewis University

Keywords: graph theory; self-assembling DNA; tiling; nanotechnology; bipartite graphs

Abstract

Motivated by the recent advancements in nanotechnology and the discovery of new laboratory techniques using the Watson-Crick complementary properties of DNA strands, formal graph theory has recently become useful in the study of self-assembling DNA complexes. Construction methods based on graph theory have resulted in significantly increased efficiency. We present the results of applying graph theoretical and linear algebra techniques for constructing crossed-prism graphs, crown graphs, book graphs, stacked book graphs, and helm graphs, along with kite, cricket, and moth graphs. In particular, we explore various design strategies for these graph families in two sets of laboratory constraints.

Author Biographies

Eric Redmon, Lewis University

Lewis University undergraduate student, class of 2020, primary undergraduate author, currently a graduate student at Marquette University.

Miles Mena, Lewis University

Lewis University undergraduate student, class of 2023.

Megan Vesta, Lewis University

Lewis University undergraduate, class of 2023.

Alvi Renzyl Cortes, Lewis University

Lewis University undergraduate student, class of 2019.

Lauren Gernes, Lewis University

Lewis University undergraduate student, class of 2021.

Simon Merheb, Lewis University

Lewis University undergraduate student.

Nick Soto, Lewis University

Lewis University undergraduate student, class of 2019.

Chandler Stimpert, Lewis University

Lewis University undergraduate student, class of 2019. Currently a graduate student at University of Texas Austin.

Amanda Harsy, Lewis University

Dr. Amanda Harsy is an Associate Professor of Mathematics at Lewis University, a four-year, private, Lasallian university located in Romeoville, Illinois, 35 miles southwest of Chicago. She is also part of the leadership team for the Southwest Chicago Suburbs Math Teachers’ Circle. She enjoys mentoring undergraduate research and has worked with over 35 students on projects involving math education, linear algebra, graph theory, spots analytics, and data science. Dr. Harsy is an active member of the greater mathematical community and has served as the ISMAA's Director of Private Colleges and the Secretary/Treasurer of the MAA Sports SIGMAA.