Well-Distributed Sets on Graphs

Abstract

Location theory is a topic widely researched in mathematics and computer science. The goal of this thesis will be to propose a new method for choosing vertices on a graph “optimally”, in terms of spread, by generalizing concepts from music theory using physical interpretations. The sets from music theory are call maximally even and they have nice properties that one would expect to have when dealing with sets that are spread apart. However, these sets are only defined for directed cycles, and hence we must find a way to generalize the definition of maximally even. We introduce well-distributed sets as sets of charged particles repelling one another on a graph. We first show that this is indeed an extension of maximally even, after which we analyse well-distributed sets and classify them completely for some special families of graphs.