Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 5-2015 Tropical Arithmetics and Dot Product Representations of Graphs Nicole Turner Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Mathematics Commons Recommended Citation Turner, Nicole, "Tropical Arithmetics and Dot Product Representations of Graphs" (2015). All Graduate Theses and Dissertations. 4460. https://digitalcommons.usu.edu/etd/4460 This Thesis is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact
[email protected]. TROPICAL ARITHMETICS AND DOT PRODUCT REPRESENTATIONS OF GRAPHS by Nicole Turner A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mathematics Approved: David E. Brown Brynja Kohler Major Professor Committee Member LeRoy Beasley Mark McLellan Committee Member Vice President for Research Dean of the School of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2015 ii Copyright c Nicole Turner 2015 All Rights Reserved iii ABSTRACT Tropical Arithmetics and Dot Product Representations of Graphs by Nicole Turner, Master of Science Utah State University, 2015 Major Professor: Dr. David Brown Department: Mathematics and Statistics A dot product representation (DPR) of a graph is a function that maps each vertex to a vector and two vertices are adjacent if and only if the dot product of their function values is greater than a given threshold. A tropical algebra is the antinegative semiring on IR[f1; −∞} with either minfa; bg replacing a+b and a+b replacing a·b (min-plus), or maxfa; bg replacing a + b and a + b replacing a · b (max-plus), and the symbol 1 is the additive identity in min-plus while −∞ is the additive identity in max-plus; the multiplicative identity is 0 in min-plus and in max-plus.