Regular configurations and TBR graphs

Abstract

This thesis consists of two parts: The first one is concerned with the theory and applications of regular configurations; the second one is devoted to TBR graphs. In the first part, a new approach is proposed to study regular configurations, an extremal arrangement of necklaces formed by a given number of red beads and black beads. We first show that this concept is closely related to several other concepts studied in the literature, such as balanced words, maximally even sets, and the ground states in the Kawasaki-Ising model. Then we apply regular configurations to solve the (vertex) cycle packing problem for shift digraphs, a family of Cayley digraphs. TBR is one of widely used tree rearrangement operationes, and plays an important role in heuristic algorithms for phylogenetic tree reconstruction. In the second part of this thesis we study various properties of TBR graphs, a family of graphs associated with the TBR operation. To investigate the degree distribution of the TBR graphs, we also study ð€€€-index, a concept introduced to measure the shape of trees. As an interesting by-product, we obtain a structural characterization of good trees, a well-known family of trees that generalizes the complete binary trees.