@article{davidson_hersh_2014, title={A lexicographic shellability characterization of geometric lattices}, volume={123}, ISSN={["1096-0899"]}, DOI={10.1016/j.jcta.2013.11.001}, abstractNote={Geometric lattices are characterized in this paper as those finite, atomic lattices such that every atom ordering induces a lexicographic shelling given by an edge labeling known as a minimal labeling. Equivalently, geometric lattices are shown to be exactly those finite lattices such that every ordering on the join-irreducibles induces a lexicographic shelling. This new characterization fits into a similar paradigm as McNamaraʼs characterization of supersolvable lattices as those lattices admitting a different type of lexicographic shelling, namely one in which each maximal chain is labeled with a permutation of {1,…,n}.}, number={1}, journal={JOURNAL OF COMBINATORIAL THEORY SERIES A}, author={Davidson, Ruth and Hersh, Patricia}, year={2014}, month={Apr}, pages={8–13} }
 @article{davidson_sullivant_2014, title={Distance-Based Phylogenetic Methods Around a Polytomy}, volume={11}, ISSN={["1557-9964"]}, DOI={10.1109/tcbb.2014.2309592}, abstractNote={Distance-based phylogenetic algorithms attempt to solve the NP-hard least-squares phylogeny problem by mapping an arbitrary dissimilarity map representing biological data to a tree metric. The set of all dissimilarity maps is a Euclidean space properly containing the space of all tree metrics as a polyhedral fan. Outputs of distance-based tree reconstruction algorithms such as UPGMA and neighbor-joining are points in the maximal cones in the fan. Tree metrics with polytomies lie at the intersections of maximal cones. A phylogenetic algorithm divides the space of all dissimilarity maps into regions based upon which combinatorial tree is reconstructed by the algorithm. Comparison of phylogenetic methods can be done by comparing the geometry of these regions. We use polyhedral geometry to compare the local nature of the subdivisions induced by least-squares phylogeny, UPGMA, and neighbor-joining when the true tree has a single polytomy with exactly four neighbors. Our results suggest that in some circumstances, UPGMA and neighbor-joining poorly match least-squares phylogeny.}, number={2}, journal={IEEE-ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS}, author={Davidson, Ruth and Sullivant, Seth}, year={2014}, pages={325–335} }
 @article{davidson_sullivant_2013, title={Polyhedral combinatorics of UPGMA cones}, volume={50}, ISSN={["1090-2074"]}, DOI={10.1016/j.aam.2012.10.002}, abstractNote={Distance-based methods such as UPGMA (Unweighted Pair Group Method with Arithmetic Mean) continue to play a significant role in phylogenetic research. We use polyhedral combinatorics to analyze the natural subdivision of the positive orthant induced by classifying the input vectors according to tree topologies returned by the algorithm. The partition lattice informs the study of UPGMA trees. We give a closed form for the extreme rays of UPGMA cones on n taxa, and compute the spherical volumes of the UPGMA cones for small n .}, number={2}, journal={ADVANCES IN APPLIED MATHEMATICS}, author={Davidson, Ruth and Sullivant, Seth}, year={2013}, month={Feb}, pages={327–338} }